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this entry is going to be one chapter of geometry of physics

under construction

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These notes mean to give an expository but rigorous introduction to the basic concepts of relativistic perturbative quantum field theories, specifically those that arise as the perturbative quantization of a Lagrangian field theory – such as quantum electrodynamics, quantum chromodynamics, and perturbative quantum gravity appearing in the standard model of particle physics.

For broad introduction of the idea of the topic of perturbative quantum field theory see there and see

• PhysicsForums-Insights: Introduction to Perturbative Quantum Field Theory

Here, first we consider classical field theory (or rather pre-quantum field theory), complete with BV-BRST formalism; then its deformation quantization via causal perturbation theory to perturbative quantum field theory. This mathematically rigorous (i.e. clear and precise) formulation of the traditional informal lore has come to be known as perturbative algebraic quantum field theory.

We aim to give a fully local discussion, where all structures arise on the “jet bundle over the field bundle” (introduced below) and “transgress” from there to the spaces of field histories over spacetime (discussed further below). This “Higher Prequantum Geometry” streamlines traditional constructions and serves the conceptualization in the theory. This is joint work with Igor Khavkine.

In full beauty these concepts are extremely general and powerful; but the aim here is to give a first precise idea of the subject, not a fully general account. Therefore we concentrate on the special case where spacetime is Minkowski spacetime (def. 23 below), where the field bundle (def. 34 below) is an ordinary trivial vector bundle (example 9 below) and hence the Lagrangian density (def. 60 below) is globally defined. Similarly, when considering gauge theory we consider just the special case that the gauge parameter-bundle is a trivial vector bundle and we concentrate on the case that the gauge symmetries are “closed irreducible” (def. 23 below). But we aim to organize all concepts such that the structure of their generalization to curved spacetime and non-trivial field bundles is immediate.

This comparatively simple setup already subsumes what is considered in traditional texts on the subject; it captures the established perturbative BRST-BV quantization of gauge fields coupled to fermions on curved spacetimes – which is the state of the art. Further generalization, necessary for the discussion of global topological effects, such as instanton configurations of gauge fields, will be discussed elsewhere (see at homotopical algebraic quantum field theory).

Alongside the theory we develop the concrete examples of the real scalar field, the electromagnetic field and the Dirac field:

running examples

field	field bundle	Lagrangian density	equation of motion
real scalar field	expl. 10	expl. 39	expl. 45
Dirac field	expl. 35	expl. 43	expl. 52
electromagnetic field	expl. 11	expl. 40	expl. 46
Yang-Mills field	expl. 12, expl. 13	expl. 41	expl. 47
B-field	expl. 14	expl 42	expl. 48

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field	Poisson bracket	causal propagator	Hadamard propagator	Feynman propagator
real scalar field	expl. 73, expl. 76	prop. 65	def. 107	def. 108
Dirac field	expl. 73, expl. 49	prop. 76	def. 109	def. 110

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field	gauge symmetry	local BRST complex	gauge fixing
electromagnetic field	expl. 92	expl. 79	expl. 106
Yang-Mills field	expl. 93	…	…
B-field	…	…	…

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The electromagnetic field and the Dirac field combined are the fields of quantum electrodynamics which we turn to at the end below.

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Acknowledgement

These notes profited greatly from discussions with Igor Khavkine.

Thanks also to Marco Benini, Klaus Fredenhagen, Arnold Neumaier, Kasia Rejzner for helpful discussion.

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A first idea of quantum field theory

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Geometry

The geometry of physics is differential geometry. This is the flavor of geometry which is modeled on Cartesian spaces $\mathbb{R}^n$ with smooth functions between them. Here we briefly review the basics of differential geometry on Cartesian spaces.

In principle the only background assumed of the reader here is

1. usual naive set theory (e.g. Lawvere-Rosebrugh 03);

2. the concept of the continuum: the real line $\mathbb{R}$, the plane $\mathbb{R}^2$, etc.

3. the concepts of differentiation and integration of functions on such Cartesian spaces;

hence essentially the content of multi-variable differential calculus.

We now discuss:

• Abstract coordinate systems

• Fiber bundles

• Synthetic differential geometry

• Differential forms

As we uncover Lagrangian field theory further below, we discover ever more general concepts of “space” in differential geometry, such as smooth manifolds, diffeological spaces, infinitesimal neighbourhoods, supermanifolds, Lie algebroids and super Lie ∞-algebroids. We introduce these incrementally as we go along:

more general spaces in differential geometry introduced further below

											higher differential geometry
differential geometry	smooth manifolds (def. 44)	$\hookrightarrow$	diffeological spaces (def. 35)	$\hookrightarrow$	smooth sets (def. 36)	$\hookrightarrow$	formal smooth sets (def. 40)	$\hookrightarrow$	super formal smooth sets (def. 48)	$\hookrightarrow$	super formal smooth ∞-groupoids (not needed in fully perturbative QFT)
infinitesimal geometry, Lie theory							infinitesimally thickened points (def. 39)		superpoints (def. 46)		Lie ∞-algebroids (def. 114)
											higher Lie theory
needed in QFT for:	spacetime (def. 23)		space of field histories (def. 16)				Cauchy surface (def. 87), perturbation theory (def. 84)		Dirac field (expl. 35), Pauli exclusion principle		infinitesimal gauge symmetry/BRST complex (expl. 97)

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Abstract coordinate systems

What characterizes differential geometry is that it models geometry on the continuum, namely the real line $\mathbb{R}$, together with its Cartesian products $\mathbb{R}^n$, regarded with its canonical smooth structure (def. 1 below). We may think of these Cartesian spaces $\mathbb{R}^n$ as the “abstract coordinate systems” and of the smooth functions between them as the “abstract coordinate transformations”.

We will eventually consider below much more general “smooth spaces” $X$ than just the Cartesian spaces $\mathbb{R}^n$; but all of them are going to be understood by “laying out abstract coordinate systems” inside them, in the general sense of having smooth functions $f \colon \mathbb{R}^n \to X$ mapping a Cartesian space smoothly into them. All structure on generalized smooth spaces $X$ is thereby reduced to compatible systems of structures on just Cartesian spaces, one for each smooth “probe” $f\colon \mathbb{R}^n \to X$. This is called “functorial geometry”.

Notice that the popular concept of a smooth manifold (def./prop. 44 below) is essentially that o a smooth space which locally looks just like a Cartesian space, in that there exist sufficiently many $f \colon \mathbb{R}^n \to X$ which are (open) isomorphisms onto their images. Historically it was a long process to arrive at the insight that it is wrong to fix such local coordinate identifications $f$, or to have any structure depend on such a choice. But it is useful to go one step further:

In functorial geometry we do not even focus attention on those $f \colon \mathbb{R}^n \to X$ that are isomorphisms onto their image, but consider all “probes” of $X$ by “abstract coordinate systems”. This makes differential geometry both simpler as well as more powerful. The analogous insight for algebraic geometry is due to Grothendieck 65; it was transported to differential geometry by Lawvere 67.

This allows to combine the best of two superficially disjoint worlds: On the one hand we may reduce all constructions and computations to coordinates, the way traditionally done in the physics literature; on the other hand we have full conceptial control over the coordinate-free generalized spaces analyzed thereby. What makes this work is that all coordinate-constructions are functorially considered over all abstract coordinate systems.

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Of course the composition $g \circ f$ of two smooth functions is again a smooth function.

The inclined reader may notice that this means that Cartesian spaces with smooth functions between them constitute a category (“CartSp”); but the reader not so inclined may ignore this.

For the following it is useful to think of each Cartesian space as an abstract coordinate system. We will be dealing with various generalized smooth spaces (see the table below), but they will all be characterized by a prescription for how to smoothly map abstract coordinate systems into them.

(Fiorenza-Schreiber-Stasheff 12, def. 6.3.9)

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fiber bundles

Given any context of objects and morphisms between them, such as the Cartesian spaces and smooth functions from def. 1 it is of interest to fix one object $X$ and consider other objects parameterized over it. These are called bundles (def. 4) below. For reference, we briefly discuss here the basic concepts related to bundles in the context of Cartesian spaces.

Of course the theory of bundles is mostly trivial over Cartesian spaces; it gains its main interest from its generalization to more general smooth manifolds (def./prop. 44 below). It is still worthwhile for our development to first consider the relevant concepts in this simple case first.

For more exposition see at fiber bundles in physics.

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synthetic differential geometry

Below we encounter generalizations of ordinary differential geometry that include explicit “infinitesimals” in the guise of infinitesimally thickened points, as well as “super-graded infinitesimals”, in the guise of superpoints (necessary for the description of fermion fields such as the Dirac field). As we discuss below, these structures are naturally incorporated into differential geometry in just the same way as Grothendieck introduced them into algebraic geometry (in the guise of “formal schemes”), namely in terms of formally dual rings of functions with nilpotent ideals. That this also works well for differential geometry rests on the following three basic but important properties, which say that smooth functions behave “more algebraically” than their definition might superficially suggest:

Actually all three statements in prop. 1 hold not just for Cartesian spaces, but generally for smooth manifolds (def./prop. 44 below; if only we generalize in the second statement from free modules to projective modules. However for our development here it is useful to first focus on just Cartesian spaces and then bootstrap the theory of smooth manifolds and much more from that, which we do below.

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differential forms

We introduce and discuss differential forms on Cartesian spaces.

We think of $d x^i$ as a measure for infinitesimal displacements along the $x^i$-coordinate of a Cartesian space. If we have a measure of infintesimal displacement on some $\mathbb{R}^n$ and a smooth function $f \colon \mathbb{R}^{\tilde n} \to \mathbb{R}^n$, then this induces a measure for infinitesimal displacement on $\mathbb{R}^{\tilde n}$ by sending whatever happens there first with $f$ to $\mathbb{R}^n$ and then applying the given measure there. This is captured by the following definition:

The following definition captures the idea that if $d x^i$ is a measure for displacement along the $x^i$-coordinate, and $d x^j$ a measure for displacement along the $x^j$ coordinate, then there should be a way to get a measure, to be called $d x^i \wedge d x^j$, for infinitesimal surfaces (squares) in the $x^i$-$x^j$-plane. And this should keep track of the orientation of these squares, with

being the same infinitesimal measure with orientation reversed.

Now all the constructions for differential 1-forms above extent naturally to differential n-forms:

Finally we turn to the concept of integration of differential forms (def. 15 below). First we need to say what it is that differential forms may be integrated over:

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This concludes our review of the basics of (synthetic) differential geometry on which the following development of quantum field theory is based. In the next chapter we consider spacetime and spin.

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Spacetime

Relativistic field theory takes place on spacetime.

The concept of spacetime makes sense for every dimension $p+1$ with $p \in \mathbb{N}$. The observable universe has macroscopic dimension $3+1$, but quantum field theory generally makes sense also in lower and in higher dimensions. For instance quantum field theory in dimension 0+1 is the “worldline” theory of particles, also known as quantum mechanics; while quantum field theory in dimension $\gt p+1$ may be “KK-compactified” to an “effective” field theory in dimension $p+1$ which generally looks more complicated than its higher dimensional incarnation.

However, every realistic field theory, and also most of the non-realistic field theories of interest, contain spinor fields such as the Dirac field (example 43 below) and the precise nature and behaviour of spinors does depend sensitively on spacetime dimension. In fact the theory of relativistic spinors is mathematically most natural in just the following four spacetime dimensions:

In the literature one finds these four dimensions advertized for two superficially unrelated reasons:

1. in precisely these dimensions “twistors” exist (see there);

2. in precisely these dimensions “GS-superstrings” exist (see there).

However, both these explanations have a common origin in something simpler and deeper: Spacetime in these dimensions appears from the “Pauli matrices” with entries in the real normed division algebras. (In fact it goes deeper still, but this will not concern us here.)

This we explain now, and then we use this to obtain a slick handle on spinors in these dimensions, using simple linear algebra over the four real normed division algebras. At the end (in remark 7) we give a dictionary that expresses these constructions in terms of the “two-component spinor notation” that is traditionally used in physics texts (remark 7 below).

The relation between real spin representations and division algebras, is originally due to Kugo-Townsend 82, Sudbery 84 and others. We follow the streamlined discussion in Baez-Huerta 09 and Baez-Huerta 10.

A key extra structure that the spinors impose on the underlying Cartesian space of spacetime is its causal structure, which determines which points in spacetime (“events”) are in the future or the past of other points (def. 29 below). This causal structure will turn out to tightly control the quantum field theory on spacetime in terms of the “causal additivity of the S-matrix” (prop. 91 below) and the induced “causal locality” of the algebra of quantum observables (prop. 94 below). To prepare the discussion of these constructions, we end this chapter with some basics on the causal structure of Minkowski spacetime.

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1. Real division algebras

2. Spacetime in dimensions 3, 4, 6 and 10

3. Lorentz group and Spin group

4. Spinors in dimensions 3, 4, 6 and 10

5. Causal structure

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Real division algebras

To amplify the following pattern and to fix our notation for algebra generators, recall these definitions:

One defines the following operations on these real algebras:

What actually matters for the following relation of the real normed division algebras to real spin representations is that they are also alternative algebras:

We record some basic properties of associators in alternative star-algebras that we need below:

It is immediate to check that:

The analog of the Hurwitz theorem (prop. 5) is now this:

This is due to (Zorn 30).

For the following, the key point of alternative algebras is this equivalent characterization:

This is due to Emil Artin, see for instance (Schafer 95, p. 18).

Proposition 9 is what allows to carry over a minimum of linear algebra also to the octonions such as to yield a representation of the Clifford algebra on $\mathbb{R}^{9,1}$. This happens in the proof of prop. 15 below.

So we will be looking at a fragment of linear algebra over these four normed division algebras. To that end, fix the following notation and terminology:

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Minkowski spacetime in dimensions 3,4,6 and 10

We now discover Minkowski spacetime of dimension 3,4,6 and 10, in terms of the real normed division algebras $\mathbb{K}$ from prop. 5, equivalently the real alternative division algebras from prop. 8: this is prop./def. 10 and def. 23 below.

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The identification of Minkowski spacetime (def. 23) in the exceptional dimensions with the generalized Pauli matrices (prop./def. 10) has some immediate useful implications:

(Baez-Huerta 09, prop. 5)

In fact the special linear groups of linear isometries in prop. 12 are the spin groups (def. 26 below) in these dimensions.

exceptional spinors and real normed division algebras

Lorentzian spacetime dimension	$\phantom{AA}$spin group	normed division algebra	$\,\,$ brane scan entry
$3 = 2+1$	$Spin(2,1) \simeq SL(2,\mathbb{R})$	$\phantom{A}$ $\mathbb{R}$ the real numbers	super 1-brane in 3d
$4 = 3+1$	$Spin(3,1) \simeq SL(2, \mathbb{C})$	$\phantom{A}$ $\mathbb{C}$ the complex numbers	super 2-brane in 4d
$6 = 5+1$	$Spin(5,1) \simeq SL(2, \mathbb{H})$	$\phantom{A}$ $\mathbb{H}$ the quaternions	little string
$10 = 9+1$	$Spin(9,1) {\simeq} \text{"}SL(2,\mathbb{O})\text{"}$	$\phantom{A}$ $\mathbb{O}$ the octonions	heterotic/type II string

This we explain now.

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Lorentz group and spin group

The following concept of the Clifford algebra (def. 25) of Minkowski spacetime encodes the structure of the inner product space $\mathbb{R}^{d-1,1}$ in terms of algebraic operation (“geometric algebra”), such that the action of the Lorentz group becomes represented by a conjugation action (example 7 below). In particular this means that every element of the proper orthochronous Lorentz group may be “split in half” to yield a double cover: the spin group (def. 26 below).

The key point of the Clifford algebra (def. 25) is that it realizes spacetime reflections, rotations and boosts via conjugation actions:

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Spinors in dimensions 3, 4, 6 and 10

We now discuss how real spin representations (def. 26) in spacetime dimensions 3,4, 6 and 10 are naturally induced from linear algebra over the four real alternative division algebras (prop. 5).

(Baez-Huerta 09, p. 6)

(Baez-Huerta 09, prop. 8, prop. 9).

(e.g. Dermisek I-8, Dermisek I-9)

Below we spell out the example of the Lagrangian field theory of the Dirac field in detail (example 43). For discussion of massive chiral spinor fields one also needs the following, here we just mention this for completeness:

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Causal structure

We need to consider the following concepts and constructions related to the causal structure of Minkowski spacetime $\Sigma$ (def. 23).

(Bär 14, section 1, Khavkine 14, def. 2.1)

The following lemma 1 will be key in the derivation (proof of prop. 92 below) of the causal locality of algebra of quantum observables in perturbative quantum field theory:

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This concludes our discussion of spin and spacetime. In the next chapter we consider the concept of fields on spacetime.

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Fields

A field history on a given spacetime $\Sigma$ (a history of spatial field configurations, see remark 8 below) is a quantity assigned to each point of spacetime (each event), such that this assignment varies smoothly with spacetime points. For instance an electromagnetic field history (example 11 below) is at each point of spacetime a collection of vectors that encode the direction in which a charged particle passing through that point would feel a force (the “Lorentz force”, see example 11 below).

This is readily formalized (def. 34 below): If $F$ denotes the smooth manifold of “values” that the given kind of field may take at any spacetime point, then a field history $\Phi$ is modeled as a smooth function from spacetime to this space of values:

It will be useful to unify spacetime and the space of field values (the field fiber) into a single manifold, the Cartesian product

and to think of this equipped with the projection map onto the first factor as a fiber bundle of spaces of field values over spacetime

This is then called the field bundle, which specifies the kind of values that the given field species may take at any point of spacetime. Since the space $F$ of field values is the fiber of this fiber bundle (def. 6), it is sometimes also called the field fiber. (See also at fiber bundles in physics.)

Given a field bundle $E \overset{fb}{\to}\Sigma$, then a field history is a section of that bundle (def. 5). The discussion of field theory concerns the space of all possible field histories, hence the space of sections of the field bundle (example 16 below). This is a very “large” generalized smooth space, called a diffeological space (def. 35 below).

Or rather, in the presence of fermion fields such as the Dirac field (example 35 below), the Pauli exclusion principle demands that the field bundle is a super-manifold, and that the fermionic space of field histories (example 53 below) is a super-geometric generalized smooth space: a super smooth set (def. 48 below).

This smooth structure on the space of field histories will be crucial when we discuss observables of a field theory below, because these are smooth functions on the space of field histories. In particular it is this smooth structure which allows to derive that linear observables of a free field theory are given by distributions (prop. 37) below. Among these are the point evaluation observables (delta distributions) which are traditionally denoted by the same symbol as the field histories themselves.

Hence there are these aspects of the concept of “field” in physics, which are closely related, but crucially different:

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aspects of the concept of fields

aspect	term	type	description	def.
field component	$\phi^a$, $\phi^a_{,\mu}$	$J^\infty_\Sigma(E) \to \mathbb{R}$	coordinate function on jet bundle of field bundle	def. 34, def. 54
field history	$\Phi$, $\frac{\partial \Phi}{\partial x^\mu}$	$\Sigma \to J^\infty_\Sigma(E)$	jet prolongation of section of field bundle	def. 34, def. 55
field observable	$\mathbf{\Phi}^a(x)$, $\partial_{\mu} \mathbf{\Phi}^a(x),$	$\Gamma_{\Sigma}(E) \to \mathbb{R}$	derivatives of delta-functional on space of sections	def. 71, example 60
averaging of field observable	$\alpha^\ast \mapsto \underset{\Sigma}{\int} \alpha^\ast_a(x) \mathbf{\Phi}^a(x) \, dvol_\Sigma(x)$	$\Gamma_{\Sigma,cp}(E^\ast) \to Obs(E_{scp},\mathbf{L})$	observable-valued distribution	def. 80
algebra of quantum observables	$\left( Obs(E,\mathbf{L})_{\mu c},\, \star\right)$	$\mathbb{C}Alg$	non-commutative algebra structure on field observables	def. 127, def. 132

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We now discuss these topics:

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• Field bundle

• Space of field histories

• Infinitesimal geometry

• Fermion fields and Supergeometry

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field bundles

For the time being, not to get distracted from the basic idea of quantum field theory, we will focus on the following simple special case of field bundles:

For readers familiar with the concepts of principal bundles and connections on a bundle we include the following example 13 which generalizes the Yang-Mills field over Minkowski spacetime from example 12 to the situation over general spacetimes.

The electromagnetic field (def. 11) and the Yang-Mills field (def. 12, def. 13) with differential 1-forms as field histories are the basic examples of gauge fields (we consider this in more detail below in Gauge symmetries). There are also higher gauge fields with differential n-forms as field histories:

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space of field histories

Given any field bundle, we will eventually need to regard the set of all field histories $\Gamma_\Sigma(E)$ as a “smooth set” itself, a smooth space of sections, to which constructions of differential geometry apply (such as for the discussion of observables and states below ). Notably we need to be talking about differential forms on $\Gamma_\Sigma(E)$.

However, a space of sections $\Gamma_\Sigma(E)$ does not in general carry the structure of a smooth manifold; and it carries the correct smooth structure of an infinite dimensional manifold only if $\Sigma$ is a compact space (see at manifold structure of mapping spaces). Even if it does carry infinite dimensional manifold structure, inspection shows that this is more structure than actually needed for the discussion of field theory. Namely it turns out below that all we need to know is what counts as a smooth family of sections/field histories, hence which functions of sets

from any Cartesian space $\mathbb{R}^n$ (def. 1) into $\Gamma_\Sigma(E)$ count as smooth functions, subject to some basic consistency condition on this choice.

This structure on $\Gamma_\Sigma(E)$ is called the structure of a diffeological space:

For more background on diffeological spaces see also geometry of physics -- smooth sets.

The following example 17 is included only for readers who wonder how infinite-dimensional manifolds fit in. Since we will never actually use infinite-dimensional manifold-structure, this example is may be ignored.

Once the step from smooth manifolds to diffeological spaces (def. 35) is made, characterizing the smooth structure of the space entirely by how we may probe it by mapping smooth Cartesian spaces into it, it becomes clear that the underlying set $X_s$ of a diffeological space $X$ is not actually crucial to support the concept: The space is already entirely defined structurally by the system of smooth plots it has, and the underlying set $X_s$ is recovered from these as the set of plots from the point $\mathbb{R}^0$.

This is crucial for field theory: the spaces of field histories of fermionic fields (def. 50 below) such as the Dirac field (example 53 below) do not have underlying sets of points the way diffeological spaces have. Informally, the reason is that a point is a bosonic object, while and the nature of fermionic fields is the opposite of bosonic.

But we may just as well drop the mentioning of the underlying set $X_s$ in the definition of generalized smooth spaces. By simply stripping this requirement off of def. 35 we obtain the following more general and more useful definition (still “bosonic”, though, the supergeometric version is def. 48 below):

Basing differential geometry on smooth sets is an instance of the general approach to geometry called functorial geometry or topos theory. For more background on this see at geometry of physics -- smooth sets.

First we verify that the concept of smooth sets is a consistent generalization:

Recall, for the next proposition 17, that in the definition 36 of a smooth set $X$ the sets $X(\mathbb{R}^n)$ are abstract sets which are to be thought of as would-be smooth functions “$\mathbb{R}^n \to X$”. Inside def. 36 this only makes sense in quotation marks, since inside that definition the smooth set $X$ is only being defined, so that inside that definition there is not yet an actual concept of smooth functions of the form “$\mathbb{R}^n \to X$”.

But now that the definition of smooth sets and of morphisms between them has been stated, and seeing that Cartesian space $\mathbb{R}^n$ are examples of smooth sets, by example 18, there is now an actual concept of smooth functions $\mathbb{R}^n \to X$, namely as smooth sets. For the concept of smooth sets to be consistent, it ought to be true that this a posteriori concept of smooth functions from Cartesian spaces to smooth sets coincides wth the a priori concept, hence that we “may remove the quotation marks” in the above. The following proposition says that this is indeed the case:

A key class of examples of smooth sets (def. 36) that are not diffeological spaces (def. 35) are universal smooth moduli spaces of differential forms:

The upshot is that we now have a good definition of differential forms on any diffeological space and more generally on any smooth set:

The usual operations on ordinary differential forms directly generalize plot-wise to differential forms on diffeological spaces and more generally on smooth sets:

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Infinitesimal geometry

It is crucial in field theory that we consider field histories not only over all of spacetime, but also restricted to submanifolds of spacetime. Or rather, what is actually of interest are the restrictions of the field histories to the infinitesimal neighbourhoods (example 27 below) of these submanifolds. This appears notably in the construction of phase spaces below. Moreover, fermion fields such as the Dirac field (example 35 below) take values in graded infinitesimal spaces, called super spaces (discussed below). Therefore “infinitesimal geometry”, sometimes called formal geometry (as in “formal scheme”) or synthetic differential geometry or synthetic differential supergeometry, is a central aspect of field theory.

In order to mathematically grasp what infinitesimal neighbourhoods are, we appeal to the first magic algebraic property of differential geometry from prop. 1, which says that we may recognize smooth manifolds $X$ dually in terms of their commutative algebras $C^\infty(X)$ of smooth functions on them

But since there are of course more algebras $A \in \mathbb{R}Algebras$ than arise this way from smooth manifolds, we may turn this around and try to regard any algebra $A$ as defining a would-be space, which would have $A$ as its algebra of functions.

For example an infinitesimally thickened point should be a space which is “so small” that every smooth function $f$ on it which vanishes at the origin takes values so tiny that some finite power of them is not just even more tiny, but actually vanishes:

(Kock 81, Kock 10)

The following example 21 shows that infinitesimal thickening is invisible for ordinary spaces when mapping out of these. In contrast example 22 further below shows that the morphisms into an ordinary space out of an infinitesimal space are interesting: these are tangent vectors and their higher order infinitesimal analogs.

We need to consider infinitesimally thickened spaces more general than the thickenings of just Cartesian spaces in def. 39. But just as Cartesian spaces (def. 1) serve as the local test geometries to induce the general concept of diffeological spaces and smooth sets (def. 36), so using infinitesimally thickened Cartesian spaces as test geometries immediately induces the corresponding generalization of smooth sets with infinitesimals:

(Dubuc 79)

Basing infinitesimal geometry on formal smooth sets is an instance of the general approach to geometry called functorial geometry or topos theory. For more background on this see at geometry of physics -- manifolds and orbifolds.

We have the evident generalization of example 15 to smooth geometry with infinitesimals:

While in infinitesimally thickened Cartesian spaces (def. 39) only infinitesimals to any finite order may exist, in formal smooth sets (def. 40) we may find infinitesimals to any arbitrary finite order:

This allows to grasp the restriction of field histories to the infinitesimal neighbourhood of a submanifold of spacetime, which will be crucial for the discussion of phase spaces below.

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We close the discussion of infinitesimal differential geometry by explaining how we may recover the concept of smooth manifolds inside the more general formal smooth sets (def./prop. 44 below). The key point is that the presence of infinitesimals in the theory allows an intrinsic definition of local diffeomorphisms/formally étale morphism (def. 43 and example 28 below). It is noteworthy that the only role this concept plays in the development of field theory below is that smooth manifolds admit triangulations by smooth singular simplices (def. 14) so that the concept of fiber integration of differential forms is well defined over manifolds.

(Khavkine-Schreiber 17, def. 3.1)

(Khavkine-Schreiber 17, prop. 3.2)

(Khavkine-Schreiber 17, example 3.4)

For more on smooth manifolds from the perspective of formal smooth sets see at geometry of physics -- manifolds and orbifolds.

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fermion fields and supergeometry

Field theories of interest crucially involve fermionic fields (def. 50 below), such as the Dirac field (example 35 below), which are subject to the “Pauli exclusion principle”, a key reason for the stability of matter. Mathematically this principle means that these fields have field bundles whose field fiber is not an ordinary manifold, but an odd-graded supermanifold (more on this in remark 17 and remark 27 below).

This “supergeometry” is an immediate generalization of the infinitesimal geometry above, where now the infinitesimal elements in the algebra of functions may be equipped with a grading: one speaks of superalgebra.

The “super”-terminology for something as down-to-earth as the mathematical principle behind the stability of matter may seem unfortunate. For better or worse, this terminology has become standard since the middle of the 20th century. But the concept that today is called supercommutative superalgebra was in fact first considered by Grassmann 1844 who got it right (“Ausdehnungslehre”) but apparently was too far ahead of his time and remained unappreciated.

Beware that considering supergeometry does not necessarily involve considering “supersymmetry”. Supergeometry is the geometry of fermion fields (def. 50 below), and that all matter fields in the observable universe are fermionic has been experimentally established since the Stern-Gerlach experiment in 1922. Supersymmetry, on the other hand, is a hypothetical extension of spacetime-symmetry within the context of supergeometry. Here we do not discuss supersymmetry; for details see instead at geometry of physics -- supersymmetry.

For more details on superalgebra see at geometry of physics -- superalgebra.

The following is the direct super-algebraic analog of the definition of infinitesimally thickened Cartesian spaces (def. 39):

All the differential geometry over Cartesian space that we discussed above generalizes immediately to super Cartesian spaces (def. 46) if we stricly adhere to the super sign rule which says that whenever two odd-graded elements swap places, a minus sign is picked up. In particular we have the following generalization of the de Rham complex on Cartesian spaces discussed above.

(e.g. Castellani-D’Auria-Fré 91 (II.2.106) and (II.2.109), Deligne-Freed 99, section 6)

Beware that there is also another sign rule for super differential forms used in the literature. See at signs in supergeometry for further discussion.

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It is clear now by direct analogy with the definition of formal smooth sets (def. 40) what the corresponding supergeometric generalization is. For definiteness we spell it out yet once more:

(Yetter 88)

Basing supergeometry on super formal smooth sets is an instance of the general approach to geometry called functorial geometry or topos theory. For more background on this see at geometry of physics -- supergeometry.

In direct generalization of example 15 we have:

Similarly, in direct generalization of prop. 17 we have:

We do not need to consider here supermanifolds more general than the super Cartesian spaces (def. 46). But for those readers familiar with the concept we include the following direct analog of the characterization of smooth manifolds according to def./prop. 44:

Finally we have the evident generalization of the smooth moduli space $\mathbf{\Omega}^\bullet$ of differential forms from example 19 to supergeometry

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What is crucial for the discussion of field theory is the following immediate supergeometric analog of the smooth structure on the space of field histories from example 16:

We will focus on discussing the supergeometric space of field histories (example 33) of the Dirac field (def. 35 below). This we consider below in example 35; but first we discuss now some relevant basics of general supergeometry.

Example 33 is really a special case of a general relative mapping space-construction as in example 25. This immediately generalizes also to the supergeometric context.

In direct generalization of the synthetic tangent bundle construction (example 26) to supergeometry we have

With this supergeometry in hand we finally turn to defining the Dirac field species:

We analyze the special nature of the supergeometry space of field histories of the Dirac field a little (prop. 53) below and conclude by highlighting the crucial role of supergeometry (remark 10 below)

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This concludes our discussion of the concept of fields itself. In the following chapter we consider the variational calculus of fields.

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Field variations

Given a field bundle as in def. 34 above, then we know what type of quantities the corresponding field histories assign to a given spacetime point (a given event). Among all consistent such field histories, some are to qualify as those that “may occur in reality” if we think of the field theory as a means to describe parts of the observable universe. Moreover, if the reality to be described does not exhibit “action at a distance” then admissibility of its field histories should be determined over arbitrary small spacetime regions, in fact over the infinitesimal neighbourhood of any spacetime point (remark 11 below). This means equivalently that the realized field histories should be those that satisfy a given differential equation, namely an equation between the partial derivatives of the field history at any spacetime point. This is called the equation of motion of the field theory (def. 61 below).

In order to formalize this, it is useful to first collect all the possible partial derivatives that a field history may have at any given point into one big space of “field derivatives at spacetime points”. This collection is called the jet bundle of the field bundle, given as def. 54 below.

Moving around in this space means to change the possible value of fields and their derivatives, hence to vary the fields. Accordingly variational calculus of fields is just differential calculus on the jet bundle of the field bundle, this we consider in def. 59 below.

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The coordinate functions $\phi^a_{\mu_1 \cdots \mu_k}$ on a jet bundle (def. 54) are to be thought of as partial derivatives $\frac{\partial}{\partial x^{\mu_1}} \cdots \frac{\partial}{\partial x^{\mu_k}} \Phi^a$ of components $\Phi^a$ of would-be field histories $\Phi$. The power of the jet bundle is that it allows to disentangle relations between would-be partial derivatives of field history components in themselves from consideration of actual field histories. In traditional physics texts this is often done implicitly. We may make it fully explit by the operation of jet prolongation which reads in a field history and records all its partial derivatives in the form of a section of the jet bundle:

(Khavkine-Schreiber 17, section 3.3)

So the canonical coordinates on the jet bundle are the spacetime-point-wise possible values of fields and field derivates, while the jet prolongation picks the actual collections of field derivatives that may occur for an actual field history.

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While the jet bundle (def. 54) is not finite dimensional, reflecting the fact that there are arbitrarily high orders of spacetime derivatives of a field histories, it turns out that it is only very “mildly infinite dimensional” in that smooth functions on jet bundles turn out to locally depend on only finitely many of the jet coordinates (i.e. only on a finite order of spacetime derivatives). This is the content of the following prop. 19.

This reflects the locality of Lagrangian field theory defined over jet bundles: If functions on the jet bundle could depend on infinitely many jet coordinates, then by Taylor series expansion of fields the function at one point over spacetime could in fact depend on field history values at a different point of spacetime. Such non-local dependence is ruled out by prop. 19 below.

In practice this means that the situation is very convenient:

1. Any given local Lagrangian density (which will define a field theory, we come to this in def. 60 below) will locally depend on some finite number $k$ of derivatives and may hence locally be treated as living on the ordinary manifold $J^k_\Sigma(E)$.

2. while at the same time all formulas (such as for the Euler-Lagrange equations, def. 61) work uniformly without worries about fixing a maximal order of derivatives.

(see Khavkine-Schreiber 17, section 2.2 and 3.3)

Example 36 shows that the de Rham differential (def. 12) may be encoded in terms of composing jet prolongation with a suitable function on the jet bundle. More generally, jet prolongation neatly encodes (possibly non-linear) differential operators:

(Khavkine 14, def. 2.4)

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$\,$

derivatives on jet bundle

def.	symbols	name in physics	name in mathematics
def. 47	$\; \mathbf{d}$	de Rham differential	de Rham differential
59	$\; d \coloneqq d x^\mu \frac{d}{d x^\mu}$	total spacetime derivative	horizontal derivative
59	$\; \frac{d}{d x^\mu} \coloneqq \frac{\partial}{\partial x^\mu} + \phi^a_{,\mu} \frac{\partial}{\partial \phi^a} + \cdots$	total spacetime derivative along $\partial_\mu$	horizontal derivative along $\partial_\mu$
59	$\; \delta \coloneqq \mathbf{d} - d$	variational derivative	vertical derivative
22	$\; \delta_{EL} \mathbf{L} \coloneqq \mathbf{d}\mathbf{L} + d \Theta_{BFV}$	Euler-Lagrange variation	Euler-Lagrange operator
85	$\; s_{BV}$	BV-differential	Koszul differential
97	$\; s_{BRST}$	BRST differential	Chevalley-Eilenberg differential
82	$\; s$	BV-BRST differential	Chevalley-Eilenberg-Koszul-Tate differential
122	$\; s - d$	local BV-BRST differential

$\,$

The following are less trivial properties of variational differential forms:

(e.g. Anderson 89, prop. 4.3)

We will encounter the extension of the exact sequence (40) further steps to the right below in example 50.

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This concludes our discussion of variational calculus on the jet bundle of the field bundle. In the next chapter we apply this to Lagrangian densities on the jet bundle, defining Lagrangian field theories.

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Lagrangians

Given any type of fields (def. 34), those field histories that are to be regarded as “physically realizable” (if we think of the field theory as a description of the observable universe) should satisfy some differential equation – the equation of motion – meaning that realizability of any field histories may be checked upon restricting the configuration to the infinitesimal neighbourhoods (example 27) of each spacetime point. This expresses the physical absence of “action at a distance” and is one aspect of what it means to have a local field theory. By remark 11 this means that equations of motion of a field theory are equations among the coordinates of the jet bundle of the field bundle.

For many field theories of interest, their differential equation of motion is not a random partial differential equations, but is of the special kind that exhibits the “principle of extremal action” (prop. 45 below) determined by a local Lagrangian density (def. 60 below). These are called Lagrangian field theories, and this is what we consider here.

Namely among all the variational differential forms (def. 59) two kinds stand out, namley the 0-forms in $\Omega^{0,0}_\Sigma(E)$ – the smooth functions – and the horizontal $p+1$-forms $\Omega^{p+1,0}_\Sigma(E)$ – to be called the Lagrangian densities $\mathbf{L}$ (def. 60 below) – since these occupy the two “corners” of the variational bicomplex (38). There is not much to say about the 0-forms, but the Lagrangian densities $\mathbf{L}$ do inherit special structure from their special position in the variational bicomplex:

Their variational derivative $\delta \mathbf{L}$ uniquely decomposes as

1. the Euler-Lagrange derivative $\delta_{EL} \mathbf{L}$ which is proportional to the variation of the fields (instead of their derivatives)

2. the total spacetime derivative $d \Theta_{BFV}$ of a potential $\Theta_{BFV}$ for a presymplectic current $\Omega_{BFV} \coloneqq \delta \Theta_{BFV}$.

This is prop. 22 below:

These two terms play a pivotal role in the theory: The condition that the first term vanishes on field histories is a differential equation on field histories, called the Euler-Lagrange equation of motion (def. 61 below). The space of solutions to this differential equation, called the on-shell space of field histories

has the interpretation of the space of “physically realizable field histories”. This is the key object of study in the following chapters. Often this is referred to as the space of classical field histories, indicating that this does not yet reflect the full quantum field theory.

Indeed, there is also the second term in the variational derivative of the Lagrangian density, the presymplectic current $\Theta_{BFV}$, and this implies a presymplectic structure on the on-shell space of field histories (def. 88 below) which encodes deformations of the algebra of smooth functions on $\Gamma_\Sigma(E)$. This deformation is the quantization of the field theory to an actual quantum field theory, which we discuss below.

$\,$

More generally:

(e.g. Dermisek I-9)

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The beauty of Lagrangian field theory (def. 60) is that a choice of Lagrangian density determines both the equations of motion of the fields as well as a presymplectic structure on the space of solutions to this equation (the “shell”), making it the “covariant phase space” of the theory. All this we discuss below. But in fact all this key structure of the field theory is nothing but the shadow (under “transgression of variational differential forms”, def. 82 below) of the following simple relation in the variational bicomplex:

The following fact is immediate from prop. 22, but of central importance, we futher amplify this in remark 16 below:

Many examples of interest fall into the following two special cases of prop. 22:

More generally:

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The key implication of the Euler-Lagrange form on the jet bundle is that it induces the equation of motion on the space of field histories:

For later use we record the following basic fact about the Klein-Gordon equation:

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This concludes our discussion of Lagrangian densities and their variational calculus. In the next chapter we consider the infinitesimal symmetries of Lagrangians.

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Symmetries

We have introduced the concept of Lagrangian field theories $(E,\mathbf{L})$ in terms of a field bundle $E$ equipped with a Lagrangian density $\mathbf{L}$ on its jet bundle (def. 60). Generally, given any object equipped with some structure, it is of paramount interest to determine the symmetries, hence the isomorphisms/equivalences of the object that preserve the given structure (this is the “Erlanger program”, Klein 1872).

The infinitesimal symmetries of the Lagrangian density (def. 66 below) send one field history to an infinitesimally nearby one which is “equivalent” for all purposes of field theory. Among these are the infinitesimal gauge symmetries which will be of concern below. A central theorem of variational calculus says that infinitesimal symmetries of the Lagrangian correspond to conserved currents, this is Noether's theorem I, prop. 30 below. These conserved currents constitute an extension of the Lie algebra of symmetries, called the Dickey bracket.

But in (54) we have seen that the Lagrangian density of a Lagrangian field theory is just one component, in codimension 0, of an inhomogeneous “Lepage form” which in codimension 1 is given by the presymplectic potential current $\Theta_{BFV}$ (50). (This will be conceptually elucidated, after we have introduced the local BV-complex, in example 75 below.) This means that in codimension 1 we are to consider infinitesimal on-shell symmetries of the Lepage form $\mathbf{L} + \Theta_{BFV}$. These are known as Hamiltonian vector fields (def. 70 below) and the analog of Noether's theorem I now says that these correspond to Hamiltonian differential forms. The Lie algebra of these infinitesimal symmetries is called the local Poisson bracket (prop. 36 below).

Noether theorem and Hamiltonian Noether theorem

$\,$ variational form $\,$	$\,$ symmetry $\,$	$\,$ homotopy formula $\,$	$\,$ physical quantity $\,\,\,$	$\,$ local symmetry algebra $\,$
Lagrangian density $\mathbf{L}$ (def. 60)	$\mathcal{L}_v \mathbf{L} = d \tilde J$	$d(\underset{= J_v}{\underbrace{\tilde J - \iota_v \Theta_{BFV}}}) = \iota_v \, \delta_{EL}\mathbf{L}$	conserved current $J_v$ (def. 66)	Dickey bracket
presymplectic current $\Omega_{BFV}$ (prop. 22)	$\mathcal{L}^{var}_v \Theta_{BFV} = \delta \tilde H$	$\delta(\underset{= H_v}{\underbrace{\tilde H_v - \iota_v \Theta_{BFV}}}) = \iota_v \Omega_{BFV}$	Hamiltonian form $H_v$ (def. 70)	local Poisson bracket (prop. 36)

$\,$

In Phase space below we transgress this local Poisson bracket of infinitesimal symmetries of the presymplectic potential current to the “global” Poisson bracket on the covariant phase space (def. 90 below). This is the structure which then further below leads over to the quantization (deformation quantization) of the prequantum field theory to a genuine perturbative quantum field theory. However, it will turn out that there may be an obstruction to this construction, namely the existence of special infinitesimal symmetries of the Lagrangian densities, called implicit gauge symmetries (discussed further below).

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We now discuss these topics:

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• Infinitesimal symmetries of the Lagrangian density

• Infinitesimal symmetries of the presymplectic potential current

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infinitesimal symmetries of the Lagrangian density

The concept of variation in def. 63 is very general, in that it allows to vary the field coordinates independently from the corresponding jets. This generality is necessary for discussion of symmetries of presymplectic currents in def. 70 below. But for discussion of symmetries of Lagrangian densities we are interested in explicitly varying just the field coordinates (def. 64 below) and inducing from this the corresponding variations of the field derivatives (prop. 28) below.

In order to motivate the following definition 64 of evolutionary vector fields we follow remark 12 saying that concepts in variational calculus are obtained from their analogous concepts in plain differential calculus by replacing plain bundle morphisms by morphism out of the jet bundle:

Given a fiber bundle $E \overset{fb}{\to} \Sigma$, then a vertical vector field on $E$ is a section of its vertical tangent bundle $T_\Sigma E$ (def. 6), hence is a bundle morphism of this form

The variational version replaces the vector bundle on the left with its jet bundle:

An evolutionary vector field (def. 64) describes an infinitesimal change of field values depending on, possibly, the point in spacetime and the values of the field and all its derivatives (locally to finite order, by prop. 19).

This induces a corresponding infinitesimal change of the derivatives of the fields, called the prolongation of the evolutionary vector field:

Now given an evolutionary vector field, we want to consider the flow that it induces on the space of field histories:

(Noether's theorem II is prop. 78 below.)

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Since an infinitesimal symmetry of a Lagrangian (def. 66) by definition changes the Lagrangian only up to a total spacetime derivative, and since the Euler-Lagrange equations of motion by construction depend on the Lagrangian density only up to a total spacetime derivative (prop. 22), it is plausible that and infinitesimal symmetry of the Lagrangian preserves the equations of motion (47), hence the shell (49). That this is indeed the case is the statement of prop. 33 below.

To make the proof transparent, we now first introduce the concept of the evolutionary derivative (def. 68) below and then observe that in terms of these the Euler-Lagrange derivative is in fact a derivation (prop. 31).

In (Olver 93, section 5.1, p. 288) the field dependent sections of def. 67, considered in local coordinates, are referred to as tuples of differential functions.

(Olver 93, def. 5.24)

(Olver 93, theorem 5.92) The following proof is due to Igor Khavkine.

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Now we may finally prove that an infinitesimal symmetry of the Lagrangian is also an infinitesimal symmetry of the Euler-Lagrange equations of motion:

As a corollary we obtain:

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infinitesimal symmetries of the presymplectic potential current

Evidently Noether's theorem I in variational calculus (prop. 30) is the special case for horizontal $p+1$-forms of a more general phenomenon relating symmetries of variational forms to forms that are closed up to a contraction. The same phenomenon applied instead to the presymplectic current yields the following:

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Since therefore both the conserved currents from Noether's theorem as well as the Hamiltonian differential forms are generators of infinitesimal symmetries of certain variational forms (namely of the Lagrangian density and of the presymplectic current, respectively) they form a Lie algebra. For the conserved currents this is sometimes known as the Dickey bracket Lie algebra. For the Hamiltonian forms it is the Poisson bracket Lie p+1-algebra. Since here for simplicity we are considering just vertical variations, we have just a plain Lie algebra. The transgression of this Lie algebra of Hamiltonian forms on the jet bundle to Cauchy surfaces yields a presymplectic structure on phase space, this we discuss below.

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The local Poisson bracket Lie algebra $(\Omega^{p,0}_{\Sigma,Ham}(E), [-,-]^{var})$ from prop. 36 is but the lowest stage of a higher Lie theoretic structure called the Poisson bracket Lie p-algebra. Here we will not go deeper into this higher structure (see at Higher Prequantum Geometry for more), but below we will need the following simple shadow of it:

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This concludes our discussion of general infinitesimal symmetries of a Lagrangian. We pick this up again in the discussion of Gauge symmetries below. First, in the next chapter we discuss the concept of observables in field theory.

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Observables

Given a Lagrangian field theory (def. 39), then a general observable quantity or just observable for short (def. 71 below), is a smooth function

on the on-shell space of field histories (example 16, example 33) hence a smooth “functional” of field histories. We think of this as assigning to each physically realizable field history $\Phi$ the value $A(\Phi)$ of the given quantity as exhibited by that field history. For instance concepts like “average field strength in the compact spacetime region $\mathcal{O}$” should be observables. In particular the field amplitude at spacetime point $x$ should be an observable, denoted $\mathbf{\Phi}^a(x)$.

In much of the literature on field theory, these point evaluation observables $\mathbf{\Phi}^a(x)$ (example below 60) are eventually referred to as “fields” themselves, blurring the distinction between

1. field species/field bundles $E$,

2. field histories/sections $\Phi$,

3. functions on the space of field histories $\mathbf{\Phi}^a(x)$.

In particular, the process of quantization (discussed in Quantization below) affects the third of these concepts only, in that it deforms the algebra structure on observables to a non-commutative algebra of quantum observables. For this reason the observables $\mathbf{\Phi}^a(x)$ are often referred to as quantum fields. But to understand the conceptual nature of quantum field theory it is important that the $\mathbf{\Phi}^a(x)$ are really the observables or quantum observables on the space of field histories.

fields

aspect	term	type	description	def.
field component	$\phi^a$, $\phi^a_{,\mu}$	$J^\infty_\Sigma(E) \to \mathbb{R}$	coordinate function on jet bundle of field bundle	def. 34, def. 54
field history	$\Phi$, $\frac{\partial \Phi}{\partial x^\mu}$	$\Sigma \to J^\infty_\Sigma(E)$	jet prolongation of section of field bundle	def. 34, def. 55
field observable	$\mathbf{\Phi}^a(x)$, $\partial_{\mu} \mathbf{\Phi}^a(x),$	$\Gamma_{\Sigma}(E) \to \mathbb{R}$	derivatives of delta-functional on space of sections	def. 71, example 60
averaging of field observable	$\alpha^\ast \mapsto \underset{\Sigma}{\int} \alpha^\ast_a(x) \mathbf{\Phi}^a(x) \, dvol_\Sigma(x)$	$\Gamma_{\Sigma,cp}(E^\ast) \to Obs(E_{scp},\mathbf{L})$	observable-valued distribution	def. 80
algebra of quantum observables	$\left( Obs(E,\mathbf{L})_{\mu c},\, \star\right)$	$\mathbb{C}Alg$	non-commutative algebra structure on field observables	def. 127, def. 132

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There are various further conditions on observables which we will eventually consider, forming subspaces of gauge invariant observables (def. 98), local observables (def. 83 below), Hamiltonian local observables (def. 89 below) and microcausal observables (def. 126). While in the end it is only these special kinds of observables that matter, it is useful to first consider the unconstrained concept and then consecutively characterize smaller subspaces of well-behaved observables. In fact it is useful to consider yet more generally the observables on the full space of field histories (not just the on-shell subspace), called the off-shell observables.

In the case that the field bundle is a vector bundle (example 9), the off-shell space of field histories is canonically a vector space and hence it makes sense to consider linear off-shell observables, i.e. those observables $A$ with $A(c \Phi) = c A(\Phi)$ and $A(\Phi_1 + \Phi_2) = A(\Phi_1) + A(\Phi_2)$. It turns out that these are precisely the compactly supported distributions in the sense of Laurent Schwartz (prop. 37 below). This fact makes powerful tools from functional analysis and microlocal analysis available for the analysis of field theory (discussed below).

More generally there are the multilinear off-shell observables, and these are analogously given by distributions of several variables (def. 76 below). In fully perturbative quantum field theory one considers only the infinitesimal neighbourhood (example 27) of a single on-shell field history and in this case all observables are in fact given by such multilinear observables (def. 84 below).

For a free field theory (def. 62) whose Euler-Lagrange equations of motion are given by a linear differential operator which behaves well in that it is “Green hyperbolic” (def. 79 below) it follows that the actual on-shell linear observables are equivalently those off-shell observables which are spatially compactly supported distributional solutions to the formally adjoint equation of motion (prop. 43 below); and this equivalence is exhibited by composition with the causal Green function (def. 78 below):

This is theorem 1 below, which is pivotal for passing from classical field theory to quantum field theory:

This fact makes, in addition, the distributional analysis of linear differential equations available for the analysis of free field theory, notably the theory of propagators, such as Feynman propagators (def. 108 below), which we turn to in Propagators below.

The functional analysis and microlocal analysis (below) of linear observables re-expressed in distribution theory via theorem 1 solves the issues that the original formulation of perturbative quantum field theory by Schwinger-Tomonaga-Feynman-Dyson in the 1940s was notorious for suffering from (Feynman 85): The normal ordered product of quantum observables in a Wick algebra of observables follows from Hörmander's criterion for the product of distributions to be well-defined (this we discuss in Free quantum fields below) and the renormalization freedom in the construction of the S-matrix is governed by the mechanism of extensions of distributions (this we discuss in Renormalization below).

Among the polynomial on-shell observables characterized this way, the focus is furthermore on the local observables:

In local field theory the idea is that both the equations of motion as well as the observations are fully determined by their restriction to infinitesimal neighbourhoods of spacetime points (events). For the equations of motion this means that they are partial differential equations as we have seen above. For the observables it should mean that they must be averages over regions of spacetime of functions of the value of the field histories and their derivatives at any point of spacetime. Now a “smooth function of the value of the field histories and their derivatives at any point” is precisely a smooth function on the jet bundle of the field bundle (example 54) pulled back via jet prolongation (def. 55). If this is to be averaged over spacetime it needs to be the coefficient of a horizontal $p+1$-form (prop. 59).

In mathematical terminology these desiderata say that the local observables in a local field theory should be precisely the “transgressions” (def. 82 below) of horizontal variational $p+1$-forms (with compact spacetime support, def. 81 below) to the space of field histories (example 16). This is def. 83 below.

A key example of a local observable in Lagrangian field theory (def. 60) is the action functional (example 66 below). This is the transgression of the Lagrangian density itself, or rather of its product with an “adiabatic switching function” that localizes its support in a compact spacetime region. In typical cases the physical quantity whose observation is represented by the action functional is the difference of the kinetic energy-momentum minus the potential energy of a field history averaged over the given region of spacetime.

The equations of motion of a Lagrangian field theory say that those field histories are physically realized which are critical points of this action functional observable. This is the principle of extremal action (prop. 45 below).

This formalizes what it means for a field history $\Phi$ to be “realizable” (physically admissible) (a solution to the Euler-Lagrange equations, def. 61) and what the (local) observable quantities on field histories are (def. 83). It remains to formalize what it means for the physical system to be in some definite state so that the observable quantities take some definite value, reflecting the properties of that state.

Whatever formalization for states of a field theory one considers, at the very least the space of states $States$ should come with a pairing linear map

which reads in an observable quantity $A$ and a state, to be denoted $\langle - \rangle$, and produces the complex number $\langle A \rangle$ which is the “value of the observable quantity $A$ in the case that the physical system is in the state $\langle -\rangle$”.

One might imagine that it is fundamentally possible to pinpoint the exact field history that the physical system is found in. From this perspective, fixing a state should simply mean to pick such a field history, namely an element $\Phi \in \Gamma_{\Sigma}(E)_{\delta_{EL}\mathbf{L} = 0}$ in the on-shell space of field histories. If we write $\langle -\rangle_{\Phi}$ for this state, its pairing map with the observables would simply be evaluation of the observable, being a function on the field history space, on that particular element in this space:

However, in the practice of experiment a field history can never be known precisely, without remaining uncertainty. Moreover, quantum physics (to which we finally come below), suggests that this is true not just in practice, but even in principle. Therefore we should allow states to be a kind of probability distributions on the space of field histories, and regard the pairing $\langle A \rangle$ of a state $\langle - \rangle$ with an observable $A$ as a kind of expectation value of the function $A$ averaged with respect to this probability distribution. Specifically, if the observable quantity $A$ is (a smooth approximation to) a characteristic function of a subset $S \subset \Gamma_{\Sigma}(E)_{\delta_{EL}\mathbf{L} = 0}$ of the space of field histories, then its value in a given state should be the probability to find the physical system in that subset of field histories.

But, moreover, the superposition principle of quantum physics says that the actually observable observables are only those of the form $A^\ast A$ (for $A^\ast$ the image under the star-operation on the star algebra of observables.

This finally leads to the definition of states in def. 86 below.

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We now discuss these topics:

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• General observables

• Polynomial off-shell observables and Distributions

• Polynomial on-shell observables and Distributional solutions to PDEs

• Local observables and Transgression

• Infinitesimal observables

• States

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General observables

The most basic kind of observables are the following:

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Polynomial off-shell Observables and Distributions

We consider here linear observables (def. 72 below) and more generally quadratic observables (def. 75) and generally polynomial observables (def. 76 below) for free field theories and discuss how these are equivalently given by integration against generalized functions called distributions (prop. 37 and prop. 38 below).

This is the basis for the discussion of quantum observables for free field theories further below.

$\,$

For proof see at distributions are the smooth linear functionals, this prop.

The identification from prop. 37 of linear off-shell observables with compactly supported distributions makes available powerful tools from functional analysis. The key fact is the following:

(e. g. Hörmander 90, theorem 4.1.5)

Proposition 38 with prop. 37 implies that with due care we may think of all linear off-shell observables as arising from integration of field histories against some “generalized smooth functions” (namely a limit of actual smooth functions):

These considerations generalize from the field bundle of the real scalar field to general field bundles (def. 34) as long as they are smooth vector bundles (def. 7):

(Bär 14, 2.1)

(Sanders 13, Bär 14)

As before in prop. 38 the actual smooth sections yield examples of distributional sections, and all distributional sections arise as limits of integrations against smooth sections:

(Bär 14, lemma 2.15)

(Sanders 13, thm. 4.3, Bär 14, lem. 2.14)

The concept of linear observables naturally generalizes to that of multilinear observables:

In summary, the above establishes that the Schwartz theory of (compactly supported) distributions neatly applies to characterize smooth polynomial observables on the diffeological space of field histories for a field bundle which is a vector bundle.

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Polynomial on-shell Observables and Distributional solutions to PDEs

While every off-shell observable induces an on-shell observables simply by restriction (78), different off-shell observables may restrict to the same on-shell observale. It is therefore useful to find a condition on off-shell observables that makes them equivalent to on-shell observables under restriction. Here we discuss in the case of sufficiently well behaved free field equations of motion – namely Green hyperbolic differential equations, def. 79 below – that this on-shell condition on the linear off-shell observables (def. 72) is that they are distributional solutions to the formal adjoint to the equations of motion, under their identification with distributions via prop. 37.

While in general the equations of motion are not Green hyperbolic – namely not in the presence of implicit infinitesimal gauge symmetries discussed in Gauge symmetries below – it turns out that up to a suitable notion of equivalence they are equivalent to those that are, this we discuss in Gauge fixing below.

$\,$

(e.g. Bär 14, def. 3.2, cor. 3.10)

(Bär 14, def. 3.2, Khavkine 14, def. 2.2)

The two archtypical examples of Green hyperbolic differential equations are the Klein-Gordon equation and the Dirac equation on Minkowski spacetime. For the moment we just cite the existence of the advanced and retarded Green functions for these, we will work these out in detail below in Propagators.

(e. g. Bär-Ginoux-Pfaeffle 07, Bär 14, example 3.3)

(Bär 14, corollary 3.15, example 3.16)

(Khavkine 14, (24), (25))

We did not require that the advanced and retarded Green functions of a Green hyperbolic differential operator are unique; in fact this is automatic:

(Bär 14, cor. 3.12

Moreover we did not require that the advanced and retarded Green functions of a Green hyperbolic differential operator come from integral kernels (“propagators”). This, too, is automatic:

(Bär 14, thm. 3.8, cor. 3.11)

We now come to the main theorem on polynomial observables:

This is due to Igor Khavkine, based on (Khavkine 14, prop. 2.1); for proof see at Green hyperbolic differential operator this lemma.

This follows from the exact sequence in lemma 3. For details of the proof see at Green hyperbolic differential operator this prop., due to Igor Khavkine.

In conclusion we have found the following:

We will be interested in those linear observables which under the identification from theorem 1 correspond to the non-singular distributions (because on these the Poisson-Peierls bracket of the theory is defined, theorem 2 below):

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Local observables

We now discuss the sub-class of those observables which are “local”.

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Infinitesimal observables

The definition of observables in def. 71 and specifically of local observables in def. 83 uses explicit restriction to the shell, hence, by the principle of extremal action (prop. 45) to the “critical locus” of the action functional. Such critical loci are often hard to handle explicitly. It helps to consider a “homological resolution” that is given, in good circumstances, by the corresponding “derived critical locus”. These we consider in detail below in Reduced phase space. In order to have good control over these resolutions, we here consider the first perturbative aspect of field theory, namely we consider the restriction of local observables to just an infinitesimal neighbourhood of a background on-shell field history:

The space of local observables in def. 84 is the quotient of a formal power series algebra by the components of the Euler-Lagrange form and by the image of the horizontal spacetime de Rham differential. It is convenient to also conceive of the components of the Euler-Lagrange form as the image of a differential, for then the algebra of local observables obtaines a cohomological interpretation, which will lend itself to computation. This differential, whose image is the components of the Euler-Lagrange form, is called the BV-differential. We introduce this now first (def. 85 below) in a direct ad-hoc way. Further below we discuss the conceptual nature of this differential as part of the construction of the reduced phase space as a derived critical locus (example 101 below).

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States

Below we consider quantum states. These are defined formally in just the same way, only that now the algebra of observables is equipped with another product, which changes the meaning of the product expression $A^\ast A$.

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This concludes our discussion of observables. In the next chapter we consider the construction of the covariant phase space and of the Poisson-Peierls bracket on observables.

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Phase space

It might seem that with the construction of the local observables (def. 83) on the on-shell space of field histories (prop. 22) the field theory defined by a Lagrangian density (def. 60) has been completely analyzed: This data specifies, in principle, which field histories are realized, and which observable properties these have.

In particular, if the Euler-Lagrange equations of motion (def. 61) admit Cauchy surfaces (def. 87 below), i.e. spatial codimension 1 slices of spacetimes such that a field history is uniquely specified already by its restriction to the infinitesimal neighbourhood of that spatial slice, then a sufficiently complete collection of local observables whose spacetime support (def. 81) covers that Cauchy surface allows to predict the evolution of the field histories through time from that Cauchy surface.

This is all what one might think a theory of physical fields should accomplish, and in fact this is essentially all that was thought to be required of a theory of nature from about Isaac Newton’s time to about Max Planck’s time.

But we have seen that a remarkable aspect of Lagrangian field theory is that the de Rham differential of the local Lagrangian density $\mathbf{L}$ (def. 60) decomposes into two kinds of variational differential forms (prop. 22), one of which is the Euler-Lagrange form which determines the equations of motion (47).

However, there is a second contribution: The presymplectic current $\Omega_{BFV} \in \Omega^{p,2}_{\Sigma}(E)$ (52). Since this is of horizontal degree $p$, its transgression (def. 82) implies a further structure on the space of field histories restricted to spacetime submanifolds of dimension $p$ (i.e. of spacetime “codimension 1”). There may be such submanifolds such that this restriction to their infinitesimal neighbourhood (example 27) does not actually change the on-shell space of field histories, these are called the Cauchy surfaces (def. 87 below).

By the Hamiltonian Noether theorem (prop. 35) the presymplectic current induces infinitesimal symmetries acting on field histories and local observables, given by the local Poisson bracket (prop. 36). The transgression (def. 82) of the presymplectic current to these Cauchy surfaces yields the corresponding infinitesimal symmetry group acting on the on-shell field histories, whose Lie bracket is the Poisson bracket pairing on on-shell observables (example 70 below). This data, the on-shell space of field histories on the infinitesimal neighbourhood of a Cauchy surface equipped with infinitesimal symmetry exhibited by the Poisson bracket is called the phase space of the theory (def. 88) below.

In fact if enough Cauchy surfaces exist, then the presymplectic forms associated with any one choice turn out do agree after pullback to the full on-shell space of field histories, exhibiting this as the covariant phase space of the theory (prop. 46 below) which is hence manifestly independent of aa choice of space/time splitting. Accordingly, also the Poisson bracket on on-shell observables exists in a covariant form; for free field theories with Green hyperbolic equations of motion (def. 79) this is called the Peierls-Poisson bracket (theorem 2 below). The integral kernel for this Peierls-Poisson bracket is called the causal propagator (prop. 42). Its “normal ordered” or “positive frequency component”, called the Hadamard propagator (def. 107 below) as well as the corresponding time-ordered variant, called the Feynman propagator (def. 108 below), which we discuss in detail in Propagators below, control the causal perturbation theory for constructing perturbative quantum field theory by deforming the commutative pointwise product of on-shell observables to a non-commutative product governed to first order by the Peierls-Poisson bracket.

To see how such a deformation quantization comes about conceptually from the phase space strucure, notice from the basic principles of homotopy theory that given any structure on a space which is invariant with respect to a symmetry group acting on the space (here: the presymplectic current) then the true structure at hand is the homotopy quotient of that space by that symmetry group. We will explain this further below. This here just to point out that the homotopy quotient of the phase space by the infinitesimal symmetries of the presymplectic current is called the symplectic groupoid and that the true algebra of observables is hence the (polarized) convolution algebra of functions on this groupoid. This turns out to the “algebra of quantum observables” and the passage from the naive local observables on presymplectic phase space to this non-commutative algebra of functions on its homotopy quotient to the symplectic groupoid is called quantization. This we discuss in much detail below; for the moment this is just to motivate why the covariant phase space is the crucial construction to be extracted from a Lagrangian field theory.

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We now discuss these topics:

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• Covariant phase space

• BV-Resolution of the covariant phase space

• Hamiltonian local observables

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Covariant phase space

(Khavkine 14, lemma 2.5)

For the free electromagnetic field (example 40) the assumptions of theorem 2 are violated, the covariant phase space does not exist. But in the discuss of Gauge fixing, below, we will find that for an equivalent re-incarnation of the electromagnetic field, they are met after all.

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BV-resolution of the covariant phase space

So far we have discussed the covariant phase space (prop. 46) in terms of explicit restriction to the shell. We now turn to the more flexible perspective where a homological resolution of the shell in terms of “antifields” is used (def. 85).

=–

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Hamiltonian local observables

We have defined the local observables (def. 83) as the transgressions of horizontal $p+1$-forms (with compact spacetime support) to the on-shell space of field histories $\Gamma_{\Sigma}(E)_{\delta_{EL}\mathbf{L} = 0}$ over all of spacetime $\Sigma$. More explicitly, these could be called the spacetime local observables.

But with every choice of Cauchy surface $\Sigma_p \hookrightarrow \Sigma$ (def. 87) comes another notion of local observables: those that are transgressions of horizontal $p$-forms (instead of $p+1$-forms) to the on-shell space of field histories restricted to the infinitesimal neighbourhood of that Cauchy surface (def. 42): $\Gamma_{\Sigma_p}(E)_{\delta_{EL} \mathbf{L} = 0}$. These are spatially local observables, with respect to the given choice of Cauchy surface.

Among these spatially local observables are the Hamiltonian local observables (def. 89 below) which are transgressions specifically of the Hamiltonian forms (def. 70). These inherit a transgression of the local Poisson bracket (prop. 36) to a Poisson bracket on Hamiltonian local observables (def. 90 below). This is known as the Peierls bracket (example 76 below).

Beware that the local observable $\tau_{\Sigma_p}(H)$ defined by a Hamiltonian differential form $H \in \Omega^{p,0}_{\Sigma,Ham}(E)$ as in def. 89 does in general depend not just on the choice of $H$, but also on the choice $\Sigma_p$ of the Cauchy surface. The exception are those Hamiltonian forms which are conserved currents:

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This concludes our discussion of the phase space and the Poisson-Peierls bracket for well behaved Lagrangian field theories. In the next chapter we discuss in detail the integral kernels corresponding to the Poisson-Peierls bracket for key classes of examples. These are the propagators of the theory.

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Propagators

In the previous chapter we have seen the covariant phase space (prop. 46) of sufficiently nice Lagrangian field theories, which is the on-shell space of field histories equipped with the presymplectic form transgressed from the presymplectic current of the theory; and we have seen that in good cases this induces a bilinear pairing on sufficiently well-behaved observables, called the Poisson bracket (def. 90), which reflects the infinitesimal symmetries of the presymplectic current. This Poisson bracket is of central importance for passing to actual quantum field theory, since, as we will discuss in Quantization below, it is the infinitesimal approximation to the quantization of a Lagrangian field theory.

We have moreover seen that the Poisson bracket on the covariant phase space of a free field theory with Green hyperbolic equations of motion – the Peierls-Poisson bracket – is determined by the integral kernel of the causal Green function (prop. 2). Under the identification of linear on-shell observables with off-shell observables that are generalized solutions to the equations of motion (theorem 1) the convolution with this integral kernel may be understood as propagating the values of an off-shell observable through spacetime, such as to then compare it with any other observable at any spacetime point (prop. 2). Therefore the integral kernel of the causal Green function is also called the causal propagator.

This means that for Green hyperbolic free Lagrangian field theory the Poisson bracket, and hence the infinitesimal quantization of the theory, is all encoded in the causal propagator. Therefore here we analyze the causal propagator, as well as its variant propagators, in detail.

The main tool for these computations is Fourier analysis (reviewed below) by which field histories, observables and propagators on Minkowski spacetime are decomposed as superpositions of plane waves of various frequencies, wave lengths and wave vector-direction. Using this, all propagators are exhibited as those superpositions of plane waves which satisfy the dispersion relation of the given equation of motion, relating plane wave frequency to wave length.

This way the causal propagator is naturally decomposed into its contribution from positive and from negative frequencies. The positive frequency part of the causal propagator is called the Hadamard propagator (def. 107 below). It turns out (prop. 69 below) that this is equivalently the sum of the causal propagator, which itself is skew-symmetric (cor. 2 below), with a symmetric component, or equivalently that the causal propagator is the skew-symmetrization of the Hadamard propagator. After quantization of free field theory discussed further below, we will see that the Hadamard propagator is equivalently the correlation function between two point-evaluation field observables (example 60) in a vacuum state of the field theory (a state in the sense of def. 86).

Moreover, by def. 78 the causal propagator also decomposes into its contributions with future and past support, given by the difference between the advanced and retarded propagators. These we analyze first, starting with prop. 64 below.

Combining these two decompositions of the causal propagator (positive/negative frequency as well as positive/negative time) yields one more propagator, the Feynman propagator (def. 108 below).

We will see below that the quantization of a free field theory is given by a “star product” (on observables) which is given by “exponentiating” these propagators. For that to make sense, certain pointwise products of these propagators, regarded as generalized functions (prop. 38) need to exist. But since the propagators are distributions with singularities, the existence of these products requires that certain potential “UV divergences” in their Fourier transforms are absent (“Hörmander's criterion”, prop. 58 below). These UV divergences are captured by what what is called the wave front set (def. 101 below).

The study of UV divergences of distributions via their wave front sets is called microlocal analysis and provides powerful tools for the understanding of quantum field theory. In particular the propagation of singularities theorem (prop. 59) shows that for distributional solutions (def. 77) of Euler-Lagrange equations of motion, such as the propagators, their singular support propagates itself through spacetime along the wave front set.

Using this theorem we work out the wave front sets of the propagators (prop. 75 below). Via Hörmander's criterion (prop. 58) this computation will serve to show why upon quantization the Hadamard propagator replaces the causal propagator in the construction of the Wick algebra of quantum observables of the free field theory (discussed below in Free quantum fields) and the the Feynman propagator similarly controls the quantum observables of the interacting quantum field theory (below in Feynman diagrams).

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The following table summarizes the structure of the system of propagators. (The column “as vacuum expectation value of field operators” will be discussed further below in Free quantum fields).

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propagators (i.e. integral kernels of Green functions)
for the wave operator and Klein-Gordon operator
on a globally hyperbolic spacetime such as Minkowski spacetime:

name	symbol	wave front set	as vacuum exp. value of field operators	as a product of field operators
causal propagator	$\begin{aligned}\Delta_S & = \Delta_+ - \Delta_- \end{aligned}$	$\phantom{A}\,\,\,-$	$\begin{aligned} & i \hbar \, \Delta_S(x,y) = \\ & \left\langle \;\left[\mathbf{\Phi}(x),\mathbf{\Phi}(y)\right]\; \right\rangle \end{aligned}$	Peierls-Poisson bracket
advanced propagator	$\Delta_+$		$\begin{aligned} & i \hbar \, \Delta_+(x,y) = \\ & \left\{ \array{ \left\langle \; \left[ \mathbf{\Phi}(x),\mathbf{\Phi}(y) \right] \; \right\rangle &\vert& x \geq y \\ 0 &\vert& y \geq x } \right. \end{aligned}$	future part of Peierls-Poisson bracket
retarded propagator	$\Delta_-$		$\begin{aligned} & i \hbar \, \Delta_-(x,y) = \\ & \left\{ \array{ \left\langle \; \left[\mathbf{\Phi}(x),\mathbf{\Phi}(y) \right] \; \right\rangle &\vert& y \geq x \\ 0 &\vert& x \geq y } \right. \end{aligned}$	past part of Peierls-Poisson bracket
Hadamard propagator	$\begin{aligned} \Delta_H &= \tfrac{i}{2}\left( \Delta_+ - \Delta_-\right) + H\\ & = \tfrac{i}{2}\Delta_S + H \\ & = \Delta_F - i \Delta_- \end{aligned}$		$\begin{aligned} & \hbar \, \Delta_H(x,y) \\ & = \left\langle \; \mathbf{\Phi}(x) \mathbf{\Phi}(y) \; \right\rangle \\ & = \underset{ = 0 }{\underbrace{\left\langle \; : \mathbf{\Phi}(x) \mathbf{\Phi}(y) : \; \right\rangle}} \\ & \phantom{=} + \left\langle \; \left[ \mathbf{\Phi}^{(-)}(x), \mathbf{\Phi}^{(+)}(y) \right] \; \right\rangle \end{aligned}$	positive frequency of Peierls-Poisson bracket, normal-ordered product, 2-point function $\phantom{=}$ of vacuum state $\phantom{=}$ or generally of $\phantom{=}$ Hadamard state
Dirac propagator	$\begin{aligned}\Delta_D &= \tfrac{1}{2}(\Delta_+ + \Delta_-) \end{aligned}$	$\phantom{A}\,\,\, +$		would-be time-ordered product away from coincident points
Feynman propagator	$\begin{aligned}\Delta_F & = \tfrac{i}{2}\left( \Delta_+ + \Delta_- \right) + H \\ & = i \Delta_D + H \\ & = \Delta_H + i \Delta_- \end{aligned}$		$\begin{aligned} & \hbar \, \Delta_F(x,y) \\ & = \left\langle \; T\left( \; \mathbf{\Phi}(x)\mathbf{\Phi}(y) \;\right) \; \right\rangle \\ & = \left\{ \array{ \left\langle \; \mathbf{\Phi}(x)\mathbf{\Phi}(x) \; \right\rangle &\vert& x \geq y \\ \left\langle \; \mathbf{\Phi}(y) \mathbf{\Phi}(x) \; \right\rangle &\vert& y \geq x } \right.\end{aligned}$	time-ordered product

(see also Kocic’s overview: pdf)

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We now discuss these topics:

• Background

  • Fourier analysis and Plane wave modes

  • Microlocal analysis and UV-Divergences

  • Cauchy principal values

• Propagators for the free scalar field on Minkowski spacetime

  • advanced and regarded propagators

  • causal propagator

  • Hadamard propagator

  • Feynman propagator

  • singular support and wave front sets

• Propagators for the Dirac field on Minkowski spacetime

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Fourier analysis and plane wave modes

By definition, the equations of motion of free field theories (def. 62) are linear partial differential equations and hence lend themselves to harmonic analysis, where all field histories are decomposed into superpositions of plane waves via Fourier transform. Here we briefly survey the relevant definitions and facts of Fourier analysis.

In formal duality to the harmonic analysis of the field histories themselves, also the linear observables (def. 72) on the space of field histories, hence the distributional generalized functions (prop. 37) are subject to Fourier transform of distributions (def. 96 below).

Throughout, let $n \in \mathbb{N}$ and consider the Cartesian space $\mathbb{R}^n$ of dimension $n$ (def. 1). In the application to field theory, $n = p + 1$ is the dimension of spacetime and $\mathbb{R}^n$ is either Minkowski spacetime $\mathbb{R}^{p,1}$ (def. 23) or its dual vector space, thought of as the space of wave vectors (def. 91 below). For $x = (x^\mu) \in \mathbb{R}^{p,1}$ and $k = (k_\mu) \in (\mathbb{R}^(p,1))^\ast$ we write

for the canonical pairing.

(e.g. Hörmander 90, def. 7.1.2)

(e.g. Hörmander, lemma 7.1.3)

(e.g. Hörmander, theorem 7.1.5)

(e.g Hörmander 90, lemma 7.1.3, theorem 7.1.6)

The Schwartz space of functions with rapidly decreasing partial derivatives (def. 92) serves the purpose to support the Fourier transform (def. 93) together with its inverse (prop. 94), but for many applications one needs to apply the Fourier transform to more general functions, and in fact to generalized functions in the sense of distributions (via this prop.). But with the Schwartz space in hand, this generalization is readily obtained by formal duality:

e.g. (Hörmander 90, def. 7.1.7)

(e.g. Hörmander 90, lemma 7.1.8)

(e.g. Hörmander 90, below lemma 7.1.8)

Property (114) of the ordinary Fourier transform on functions with rapidly decreasing partial derivatives motivates and justifies the fullowing generalization:

(e.g. Hörmander 90, def. 1.7.9)

(Hörmander 90, theorem 7.1.14)

We have the following distributional generalization of the basic property (113) from prop. 52:

(e.g. Hörmander 90, theorem 7.1.15)

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microlocal analysis and ultraviolet divergences

A distribution (def. 37) or generalized function (prop. 38) is like a smooth function which may have “singularities”, namely points at which it values or that of its derivatives “become infinite”. Conversely, smooth functions are the non-singular distributions (prop. 38). The collection of points around which a distribution is singular (i.e. not non-singular) is called its singular support (def. 99 below).

The Fourier transform of distributions (def. 96) decomposes a generalized function into the plane wave modes that it is made of (def. 91). The Paley-Wiener-Schwartz theorem (prop. 56 below) says that the singular nature of a compactly supported distribution may be read off from this Fourier mode decomposition: Singularities correspond to large contributions by Fourier modes of high frequency and small wavelength, hence to large “ultraviolet” (UV) contributions. Therefore the singular support of a distribution is the set of points around which the Fourier transform does not sufficiently decay “in the UV”.

But since the Fourier transform is a function of the full wave vector of the plane wave modes (def. 91), not just of the frequency/wavelength, but also of the direction of the wave vector, this means that it contains directional information about the singularities: A distribution may have UV-singularities at some point and in some wave vector direction, but maybe not in other directions.

In particular, if the distribution in question is a distributional solution to a partial differential equation (def. 77) on spacetime then the propagation of singularities theorem (prop. 59 below) says that the singular support of the solution evolves in spacetime along the direction of those wave vectors in which the Fourier transform exhibits high UV constributions. This means that these directions are the “wave front” of the distributional solution. Accordingly, the singular support of a distribution together with, over each of its points, the directions of wave vectors in which the Fourier transform around that point has large UV constributions is called the wave front set of the distribution (def. 101 below).

What is called microlocal analysis is essentially the analysis of distributions with attention to their wave front set, hence to the wave vector-directions of UV divergences.

In particular the product of distributions is well defined (only) if the wave front sets of the distributions to not “collide”. And this in fact motivates the definition of the wave front set:

To see this, let $u,v \in \mathcal{D}'(\mathbb{R}^1)$ be two distributions, for simplicity of exposition taken on the real line.

Since the product $u \cdot v$, is, if it exists, supposed to generalize the pointwise product of smooth functions, it must be fixed locally: for every point $x \in \mathbb{R}$ there ought to be a compactly supported smooth function (bump function) $b \in C^\infty_{cp}(\mathbb{R})$ with $f(x) = 1$ such that

But now $b v$ and $b u$ are both compactly supported distributions (def. 100 below), and these have the special property that their Fourier transforms $\widehat{b v}$ and $\widehat{b u}$ are, in particular, smooth functions (by the Paley-Wiener-Schwartz theorem, prop 53).

Moreover, the operation of Fourier transform interchanges pointwise products with convolution products (prop. 52). This means that if the product of distributions $u \cdot v$ exists, it must locally be given by the inverse Fourier transform of the convolution product of the Fourier transforms $\widehat {b u}$ and $\widehat b v$:

(Notice that the converse of this formula holds as a fact by prop. 55)

This shows that the product of distributions exists once there is a bump function $b$ such that the integral on the right converges as $k_{max} \to \infty$.

Now the Paley-Wiener-Schwartz theorem says more, it says that the Fourier transforms $\widehat {b u}$ and $\widehat {b u}$ are polynomially bounded. On the other hand, the integral above is well defined if the integrand decreases at least quadratically with $k \to \infty$. This means that for the convolution product to be well defined, either $\widehat {b u}$ has to polynomially decrease faster with $k \to \pm \infty$ than $\widehat {b v}$ grows in the other direction, $k \to \mp \infty$ (due to the minus sign in the argument of the second factor in the convolution product), or the other way around.

Moreover, the degree of polynomial growth of the Fourier transform increases by one with each derivative. Therefore if the product law for derivatives of distributions is to hold generally, we need that either $\widehat{b u}$ or $\widehat{b v}$ decays faster than any polynomial in the opposite of the directions in which the respective other factor does not decay.

Here the set of directions of wave vectors in which the Fourier transform of a distribution localized around any point does not decay exponentially is the wave front set of a distribution (def. 101 below). Hence the condition that the product of two distributions is well defined is that for each wave vector direction in the wave front set of one of the two distributions, the opposite direction must not be an element of the wave front set of the other distribution. This is called Hörmander's criterion (prop. 58 below).

We now say this in detail:

(Hörmander 90, around (8.1.1))

(Hörmander 90, def. 8.1.2)

(Bengel 77, prop. 3.1)

(Hörmander 71, def. 1.1.1 and first sentence of section 2.1 with (1.4.1))

(Duistermaat-Hörmander 72, theorem 6.1.1, recalled for instance as Radzikowski 96, theorem 4.6)

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Cauchy principal value

An important application of the Fourier analysis of distributions is the class of distributions known broadly as Cauchy principal values. Below we will find that these control the detailed nature of the various propagators of free field theories, notably the Feynman propagator is manifestly a Cauchy principal value (prop. 71 and def. 110 below), but also the singular support properties of the causal propagator and the Hadamard propagator are governed by Cauchy principal values (prop. 72 and prop. 73 below). This way the understanding of Cauchy principal values eventually allows us to determine the wave front set of all the propagators (prop. 75) below.

Therefore we now collect some basic definitions and facts on Cauchy principal values.

The Cauchy principal value of a function which is integrable on the complement of one point is, if it exists, the limit of the integrals of the function over subsets in the complement of this point as these integration domains tend to that point symmetrically from all sides.

One also subsumes the case that the “point” is “at infinity”, hence that the function is integrable over every bounded domain. In this case the Cauchy principal value is the limit, if it exists, of the integrals of the function over bounded domains, as their bounds tend symmetrically to infinity.

The operation of sending a compactly supported smooth function (bump function) to Cauchy principal value of its pointwise product with a function $f$ that may be singular at the origin defines a distribution, usually denoted $PV(f)$.

This follows by the characterization of extension of distributions to a point, see there at this prop. (Hörmander 90, thm. 3.2.4)

Conversely, by the Fourier inversion theorem, the Fourier transform of the Heaviside distribution is the Cauchy principal value as in prop. 61:

Let now $q \colon \mathbb{R}^{n} \to \mathbb{R}$ be a non-degenerate real quadratic form analytically continued to a real quadratic form

Write $\Delta$ for the determinant of $q$

Write $q^\ast$ for the induced quadratic form on dual vector space. Notice that $q$ (and hence $a^\ast$) are assumed non-degenerate but need not necessarily be positive or negative definite.

(Gel’fand-Shilov 66, III 2.8 (8) and (9), p 289)

(Gel’fand-Shilov 66, III 2.11 (7), p 294)

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propagators for the free scalar field on Minkowski spacetime

1. Advanced and regarded propagators

2. Causal propagator

3. Hadamard propagator

4. Feynman propagator

5. Singular support and Wave front sets

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On Minkowski spacetime $\mathbb{R}^{p,1}$ consider the Klein-Gordon operator (example 25)

By example 83 its Fourier transform is

The dispersion relation of this equation we write (see def. 91)

where on the right we choose the non-negative square root.

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advanced and retarded propagators for Klein-Gordon equation on Minkowski spacetime

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causal propagator

We consider a couple of equivalent expressions for the causal propagator which are useful for computations:

graphics grabbed from Kocic 16

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Hadamard propagator

Prop. 67 exhibits the causal propagator of the Klein-Gordon operator on Minkowski spacetime as the difference of a contribution for positive temporal angular frequency $k_0 \propto \omega(\vec k)$ (hence positive energy $\hbar \omega(\vec k)$ and a contribution of negative temporal angular frequency.

The positive frequency contribution to the causal propagator is called the Hadamard propagator (def. 107 below), also known as the the vacuum state 2-point function of the free real scalar field on Minkowski spacetime. Notice that the temporal component of the wave vector is proportional to the negative angular frequency

(see at plane wave), therefore the appearance of the step function $\Theta(-k_0)$ in (129) below:

(e.g. Khavkine-Moretti 14, equation (38) and section 3.4)

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Feynman propagator

We have seen that the positive frequency component of the causal propagator $\Delta_S$ for the Klein-Gordon equation on Minkowski spacetime (prop. 64) is the Hadamard propagator $\Delta_H$ (def. 107) given, according to prop. 69, by (131)

There is an evident variant of this combination, which will be of interest:

As before for the causal propagator, there are equivalent reformulations of the Feynman propagator which are useful for computations:

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singular support and wave front sets

We now discuss the singular support (def. 99) and the wave front sets (def. 101) of the various propagators for the Klein-Gordon equation on Minkowski spacetime.

(e.g DeWitt 03 (27.85))

(Radzikowski 96, (16))

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propagators for the Dirac equation on Minkowski spacetime

We now discuss how the propagators for the free Dirac field on Minkowski spacetime (example 64) follow directly from those for the scalar field discussed above.

Similarly we obtain the other propagators for the Dirac field from those of the real scalar field:

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This concludes our discussion of propagators induced from the covariant phase space of Green hyperbolic free Lagrangian field theory. These propagators will be the key in for quantization via causal perturbation theory. But not all free field theories have a covariant phase space of Green hyperbolic equations of motion, for instance the electromagnetic field, a priori, does not. Therefore before turning to quantization in the next chapter we first discuss how gauge symmetries obstruct the existence of Green hyperbolic equations of motion.

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Gauge symmetries

The existence of the covariant phase space (prop. 46) of a Lagrangian field theory requires the existence of Cauchy surfaces (def. 87) for its Euler-Lagrange equations of motion This the case of free field theory (def. 62) this means that the equations of motion are Green hyperbolic (def. 79).

We have seen that this is the case for instance for the scalar field (example 63) and the Dirac field (example 64), but it is not the case generally, for instance it fails for the electromagnetic field (example 46), the Yang-Mills field (example 41) and the B-field (example 42). An obstruction to the existence of the covariant phase space turns out to be (prop. 77 below) the presence of infinitesimal symmetries of the Lagrangian (def. 66) that have compact spacetime support (def. 81).

An class of examples of such are those infinitesimal symmetries of the Lagrangian which occur linearly parameterized by arbitrary sections (and their derivatives) of some vector bundle on spacetime. Because then for every choice of section of compact support the corresponding symmetry will have compact spacetime support. These parameterized infinitesimal symmetries of the Lagrangian are called infinitesimal gauge symmetries, and their parameters we call the gauge parameters (def. 111 below).

Typically all compactly supported infinitesimal symmetries of the Lagrangian arise from parameterized symmetries this way; this is notably the case for the Lagrangian density of the electromagnetic field (example 92) and more generally of the Yang-Mills field.

Therefore the presence of infinitesimal symmetries of the Lagrangian with compact spacetime support is a defect of the theory which however implies its own solution, by indicating which relations ought to be promoted to “gauge” equivalences.

This obstruction is neatly captured by the cochain cohomology of the local BV-complex (def. 85) of the Lagrangian field theory (prop. 83 below). This may be understood as the algebra of functions on an extension of the jet bundle from a (locally pro-finite dimensional, prop. 112) smooth manifold to a differential graded manifold. This appearance of homotopy theory in the guise of homological algebra in Lagrangian field theory paves the way to understanding the cause of the obstruction: It disappears when the field bundle (or more generally its jet bundle) is promoted to its infinitesimal homotopy quotient by the action of these compactly supported symmetries (the “action Lie algebroid”, def. 115 below).

Passing to this homotopy quotient means to hard-wire into the geometry of the types of field their equivalence under these symmetries: in physics this is called gauge equivalence. The result is called the “reduced phase space”, which we turn to further below.

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We now discuss these topics

• Infinitesimal gauge symmetries

• Lie algebra action and Lie algebroids

• BRST complex

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As an immediate corollary of prop. 34 we have the following important observation:

Notice that there always exist spacetime-compactly supported infinitesimal symmetries that however do vanish on-shell:

(e.g. Henneaux 90, section 2.5)

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infinitesimal gauge symmetries

Prop. 77 says that the problem is to identify the presence of spacetime-compactly supported infinitesimal symmetries that are on-shell non-trivial.

One way they may be identified is if infinitesimal symmetries appear in linearly parameterized collections, where the parameter itself is an arbitrary spacetime-dependent section of some fiber bundle (hence is itself like a field history), because then choosing the parameter to have compact support yields an infinitesimal gauge symmetry. In this case we speak of a gauge parameter (def. 111 below). It turns out that in most examples of Lagrangian field theories of interest, the infinitesimal gauge symmetries all come from gauge parameters this way, and often “gauge symmetry” is undertood by default to refer to this case. Therefore we now consider this case in detail.

The point is that infinitesimal gauge symmetries in particular yield spacetime-compactly supported infinitesimal gauge symmetries:

The following is a way to identify infinitesimal gauge symmetries:

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More generally:

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Lie algebra action and Lie algebroids

Making the implicit infinitesimal gauge symmetries explicit means to make explicit how they act on the fields. To this end consider the general concept of an action of a Lie algebra by infinitesimal diffeomorphisms:

We have seen that the presence of non-trivial implicit infinitesimal gauge transformations (def. \ref{ImplicitInfinitesimalGaugeSymmetry}) in a Lagrangian field theory obstructs the existence of the covariant phase space of the theory (prop. 77). But these implicit infinitesimal gauge symmetries become explicit by hard-wiring into the very geometry of the types of fields their equivalence under these symmetries: In physics this is called gauge equivalence.

Mathematically this means to pass to the infinitesimal homotopy quotient of the action of the gauge symmetries on the shell, represented by the action Lie algebroid (def. 115 below). This is called the local reduced phase space of the theory. Such “higher structures” exist in the unification of differential geometry with homotopy theory called higher differential geometry. The (“Chevalley-Eilenberg”-)algebra of functions on this “field bundle with infinitesimal gauge symmetries made explicit” is called the BRST complex. In this cochain complex the formerly implicit infinitesimal gauge symmetries appear explicitly in the guise of field variables of positive (i.e. “higher”) degree in a differential graded-commutative algebra. These are called ghost fields.

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The two basic examples 95 are unified by the concept of action Lie algebroid (def. 115 below), which is the one of central relevance for the discussion of gauge theory: the local BRST complex (def. 97 below).

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local off-shell BRST complex

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This concludes our discussion of gauge symmetries as such. In the next chapter we discuss the homotopy quotient of the covariant phase space by the gauge symmetries, called the reduced phase space.

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Reduced phase space

We have seen above that the covariant phase space of a Lagrangian field theory is, if it exists, the “covariant transgression” of the shell (prop. 22) equipped with the local Poisson bracket (example 36). The local observables (def. 83) which operationally define the classical field theory (or rather prequantum field theory) are functions on the covariant phase space, and the local Poisson bracket on them operationally defines the corresponding quantum field theory (below). Therefore the existence of the covariant phase space is crucial for the construction of the field theory.

However, we have then seen in prop. 77 that there may be an obstruction to the existence of the covariant phase space, namely the presence of infinitesimal gauge symmetries of the Lagrangian which have been “left implicit”. We have then discussed how to make these infinitesimal gauge symmetries “explicit” by hard-wiring their action into the geometry of the fields by passing to the corresponding infinitesimal homotopy quotient (def. 114) of (the jet bundle of) the field bundle, given by the corresponding action Lie algebroid (def. 115). Its Chevalley-Eilenberg algebra of functions is called the local BRST complex of the theory (example 97).

The corresponding covariant phase space with infinitesimal gauge symmetries made explicit is now correspondingly given by the shell (48) not inside the plain space of fields, but inside this homotopy quotient by the infinitesimal gauge symmetries. This homotopy quotient of the naive phase space by the infinitesimal gauge symmetries is called the reduced phase space. Refined to its local incarnation in the jet bundle we may call this the “derived reduced prolonged shell” (def. 120 below). Its algebra of functions is called the local BV-BRST complex of the theory.

In the next section below we find that, at least in good situations, if all non-trivial implicit infinitesimal gauge symmetries have been made explicit this way by hard-wiring their action into the geometry of the reduced phase space, then the obstruction to the existence of the covariant phase space vanishes. Hence in this case then the (perturbative) quantum field theory can exist (discussed further below). This is why we do need to pass to the reduced phase space.

In order to exhibit the key structure of the reduced phase space without getting distracted by the local jet bundle geometry, we first discuss now the simple form in which it would appear after transgression (def. 82) if spacetime were compact, so that, by the principle of extremal action (prop. 45), it would be the derived critical locus ($d S = 0$) of a globally defined action functional $S$. This is example 100 below.

This serves as a warmup to the true construction of the derived shell in the action Lie algebroid of the jet bundle, where the action functional is “de-transgressed” to the Lagrangian density, which is invariant under gauge transformations only up to horizontally exact terms. This culminates in example 82 below.

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The key to understanding the “derived reduced prolonged shell”, and hence the reduced phase space, as a derived critical locus is first to exhibit the Euler-Lagrange variation of the action functional, or rather of the Lagrangian density, as a section of the analog of a cotangent bundle, but now in the realm of Lie ∞-algebroids (prop. 80 and prop. 81 below). To this end we need to first of all consider homomorphisms of Lie algebroids:

Given a gauge invariant function, hence a function $S \colon X/\mathfrak{g} \to \mathbb{R}$ on a Lie algebroid (example 98), its exterior derivative $d S$ should be a section of the cotangent bundle of the Lie algebroid. Moreover, if all field variations are infinitesimal (as in def. 84) then it should in fact be a section of the infinitesimal neighbourhood (example 27) of the zero section inside the cotangent bundle, the infinitesimal cotangent bundle $T^\ast(X/\mathfrak{g})$ of the Lie algebroid (def. 117 ebelow).

To motivate the definition 117 below of infinitesimal cotangent bundle of a Lie algebroid recall from example 27 that the algebra of functions on the infinitesimal cotangent bundle should be fiberwise the formal power series algebra in the linear functions. But a fiberwise linear function on a cotangent bundle is by definition a vector field. Finally observe that vector fields are equivalently derivations of smooth functions (prop. 1). This leads to the following definition:

In applications of interest, the spacetime $\Sigma$ is not compact. In that case one may still appeal to a construction on the space of field histories as in example 100 by considering the action functional for all adiabatically switched $b \mathbf{L}$ Lagrangians, with $b \in C_{cp}^\infty(\Sigma)$. This approach is taken in (Fredenhagen-Rejzner 11a).

Here we instead consider now the “local lift” or “de-transgression” of the above construction from the space of field histories to the jet bundle of the field bundle of the theory, refining the BV-BRST complex to the local BV-BRST complex, corresponding to the local BRST complex from example 97.

This requires a slight refinement of the construction that leads to example 100: In contrast to the action functional $S = \tau_\Sigma(\mathbf{L})$, the Lagrangian density $\mathbf{L}$ is not strictly invariant under implicit infinitesimal gauge transformations, in general, rather it may change up to a horizontally exact term (by the very definition \ref{ImplicitInfinitesimalGaugeSymmetry}). The same is then true for its Euler-Lagrange variational derivative $\delta_{EL} \mathbf{L}$. This means that $\delta_{EL} \mathbf{L}$ is not a section of the infinitesimal cotangent bundle (def. 117) of the gauge action Lie algebroid on the jet bundle, but by a local version of it, which is twisted by horizontally exact terms.

The following definition 119 is the local refinement of def. 117:

The local refinement of prop. 80 is now this:

The local analog of def. 118 is now the following definition 120 of the “derived prolonged shell” of the theory (recall the ordinary prolonged shell $\mathcal{E}^\infty \hookrightarrow J^\infty_\Sigma(E)$ from (49)):

The local refinement of example 100 is now the following:

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So far the discussion yields just the algebra of functions on the derived reduced prolonged shell. We now discuss the derived analog of the full variational bicomplex (def. 59) to the derived reduced shell.

The analog of the de Rham complex of a derived Lie algebroid is called the Weil algebra:

smooth manifolds	derived Lie algebroids
algebra of functions	Chevalley-Eilenberg algebra
algebra of differential forms	Weil algebra

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It turns out that the local BV-BRST cohomology (prop. 82) of the “derived reduced prolonged shell” very neatly captures all the aspects of Lagrangian field theory that we have been discussing so far:

(…)

Gauge fixing

We had seen above that the key intermediate construction for obtaining the quantum field theory induced from a Lagrangian density is its covariant phase space (prop. 46). But then we have seen that there are generically obstructions to the existence of the covariant phase space, embodied by infinitesimal gauge transformation that have been “left implicit” (prop. 77). We have then discussed the reduced phase space above which makes the infinitesimal gauge symmetries “explicit” by forming their homotopy quotient, whose algebra of functions, on the derived shell, is the BV-BRST complex of the theory. It remains to show that this construction of the reduced phase space indeed serves to lift the obstruction to the existence of the covariant phase space. This is the topic of gauge fixing (def. … below).

The point is that while the reduced phase space reflected by the BV-BRST complex may still not be manifestly covariant, its existence as an object in homotopy theory, here specifically in homological algebra, means that it comes with a more flexible concept of “equality”, namely homotopy equivalence, which here specifically means quasi-isomorphism.

Broadly speaking, the gauge principle in physics, says that no two things (field histories, etc.) are ever really equal, instead they may be connected by gauge transformations, and the mathematical reflection of that is the principle of homotopy theory, where no two homotopy types are ever equal, instead they may be connected by (weak) homotopy equivalence. Hence picking a specific representative of a homotopy type means to fix a gauge.

Concretely, let $(E,\mathbf{L})$ be a Lagrangian field theory with closed irreducibe gauge parameter bundle $\mathcal{G}$ and let $BV(s)$ be the corresponding BV-BRST complex (prop. 82).

We then ask for another field bundle $E^g$, possibly itself already a graded manifold, hence an object in higher differential geometry, and then we ask for a Lagrangian density $\mathbf{L}^g$ that may also genuinely live in higher prequantum geometry, hence which is defined right away on the action Lie algebroid $E^g(\mathcal{G} \times_\Sigma T_\Sigma)$ (example 97) not necessarily descending to there from $E^g$ itself. We may still form the local derived critical locus of $\mathbf{L}^g$ in $E^g(\mathcal{G} \times_\Sigma T \Sigma)$ and obtain a corresponding BV-BRST-like complex $BV(s^g)$.

We ask now that $BV(s^g)$ has particularly good properties:

1. We ask that the Koszul-Tate component $s^g_{BV}$ of $s^g$ has vanishing cochain cohomology in negative degree, which means by prop. 83 that the Lagrnagian $\mathbf{L}^g$ on the graded field bundle $E^g$ is degreewise free of the obstruction to the existence of a covariant phase space.

2. We ask moreover that the remaining Chevalley-Eilenberg component $s^g_{BRST}$ of $s^g$ is compatible with the graded Poisson bracket of this graded covariant phase space

This means that the “dg-Lagrangian field theory ”$(E^g, \mathbf{L}^g)$ induces a covariant reduced phase space “internal to” dg-manifolds; hence a “dg-covariant reduced phase space”: a graded covariant reduced phase space equipped with compatible differentials.

Such derived phase spaces $BV(s^g)$ are amenable to degreewise quantization (discussed below) if only one can keep the degreewise quantization compatible with the differential. This may be shown (below…) to be the case, and hence performing the quantization degreewise and passing in the end to the cochain cohomology of the resulting BV-BRST complex of quantum observables yields the gauge invariant local observables of the quantum field theory. This is called the “BV-BRST quantization of gauge theories”.

In order to apply this to the Lagrangian field theory $(E,\mathbf{L})$ that we actually started out to consider, we now only need to ensure that the “manifestly covariant” dg-Lagrangian field theory $(E^g, \mathbf{L}^g)$ is not necessarily equal to $(E,\mathbf{L})$, but homotopy equivalent to it, as an object in higher prequantum geometry, hence that there is a quasi-isomorphism between the corresponding BV-BRST complexes

The choice of this quasi-isomorphism hence means a choice of particularly good (namely manifestly covariant) representative of the homotopy type of $BV(s)$, and hence this is called a gauge fixing of $BV(s)$.

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Here:

term	meaning
“phase space”	derived critical locus of Lagrangian equipped with Poisson bracket
“reduced”	gauge transformations have been homotopy-quotiented out
“covariant”	Cauchy surfaces exist degreewise

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In practice this choice of gauge fixing by choice of quasi-isomorphism to a “manifestly covariant” BV-BRST complex is realized as the composite of two seperate quasi-isomorphisms:

1. an “anti-canonical transformation”

   $$e^{-\{\psi,-\}} \;\colon\; BV(s^g) \longrightarrow BV( \exp^{-\{\psi,-\}} s^g )$$

   (induced by a degree -1 element $\psi$ called, for better or worse, the “gauge fixing fermion”) which is actually a genuine isomorphism, not just a quasi-isomorphism;

2. a genuine quasi-isomorphism which contracts away a contractible direct summand of auxiliary fields

   $$BV( e^{-\{\psi,-\}} s^g ) \overset{ \text{forget contractible complex} \atop \text{of auxiliary fields} }{\longrightarrow} BV(s) .$$

(…)

(…)

Proposition 77 implies that we need a good handle on determining whether the space of implicit infinitesimal gauge symmetries modulo trivial ones is non-zero. This obstruction turns out to be neatly captured by methods of homological algebra applied to the local BV-complex (def. 85):

Prop. 83 says what gauge fixing has to accomplish: given a local BV-BRST complex we need to find a quasi-isomorphism to another complex which is such that it comes from a graded Lagrangian density whose BV-cohomology vanishes in degree -1 and hence induces a graded covariant phase space, and such that the remaining BRST differential respects the Poisson bracket on this graded covariant phase space.

(…)

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Quantization

Given any space with infinitesimal symmetries acting on it, there is the corresponding homotopy quotient by these infinitesimal symmetries. For the covariant phase space of a Lagrangian field theory, as above with its Poisson Lie algebra of infinitesimal symmetries (def. 90), this infinitesimal homotopy quotient is known as the Poisson Lie algebroid and the corresponding genuine homotopy quotient is known as the symplectic groupoid. As one passes from phase space to its symplectic groupoid, the algebra of functions on phase space – hence the algebra of observables $Obs$ (def. 83) – which is always a commutative algebra deforms to the corresponding algebra of functions on a Lie groupoid called the (polarized) convolution algebra of a Lie groupoid. This is now a non-commutative algebra called the algebra of quantum observables $Obs_{\hbar}$; and this passage from phase space to its symplectic groupoid homotopy quotient by Hamiltonian symmetries is called quantization (specifically: “geometric quantization of symplectic groupoids”). Here the strength of the non-commutativity is measured by a deformation parameter called Planck's constant $\hbar$.

Since the product in the algebra of quantum observables $Obs_\hbar$ differs from that in $Obs$, the positivity condition $\langle A^\ast A\rangle \geq 0$ in the definition of states $\langle - \rangle$ of a field theory (def. 86) acquires a different meaning. The states after quantization are called quantum states and their difference witnesses that after quantization the Lagrangian field theory is of a different nature: one says that it is no longer a classical field theory, but a quantum field theory and that the objects whose states are expressed by these new quantum states are quantum fields.

Unfortunately, explicitly constructing the algebra of quantum observables of a Lagrangian field theory and hence “constructing the quantum field theory” turns out to be extremely hard, unless some simplifying assumptions are made.

One kind of simplification occurs when the spacetime dimension is very low. For instance if the spacetime dimension is taken to be $p+1 = 1$ – modelling the approximation where one completely ignores the variation of fields in space and retains just their time evolution – then one speaks of quantum mechanics, which is well understood. Another simplification occurs when the field theory is a free field theory, meaning that its equation of motion is a normally hyperbolic linear differential operator. In this case the quantum field theory is fully understood as long as the underlying spacetime is a time-orientable and globally hyperbolic. But, as the name indicates, this captures only the case where there is no interaction among the fields.

Since the algebra of quantum observables $Obs_\hbar$ is a deformation with strength $\hbar$ of the commutative algebra of classical observables $Obs$ controlled by the Poisson Lie algebra, another simplification occurs if one gives up on the demand to understand the full deformation at finite value of Planck's constant $\hbar$ and considers just infinitesimal values of $\hbar$. Since this means that the resulting quantum observables are no longer actual smooth functions of $\hbar$, but just formal power series, this is called formal deformation quantization. The resulting “infinitesimally quantized” field theory is called perturbative quantum field theory.

For interacting field theories in spacetime dimension $p+1 \geq 3+1$ their quantization has been constructed to date only in perturbation theory this way. The construction of full non-perturbative quantum field theory (in dimension $\geq 3+1$ with non-vanishing interaction) is, at the time of this writing, a wide open problem.

But perturbative quantum field theory is well understood. This we turn to next…

(…)

Let $V$ be a vector space of finite dimension and let $\omega \in V \otimes V$ be an element of the tensor product (not necessarily skew symmetric at the moment).

We may canonically regard $V$ as a smooth manifold, in which case $\omega$ is canonically regarded as a constant rank-2 tensor. As such it has a canonical action by forming derivatives on the tensor product of the space of smooth functions:

If $\{\partial_i\}$ is a linear basis for $V$, identified, as before, with a basis for $\Gamma(T V)$, then in this basis this operation reads

where $\partial_i f \coloneqq \frac{\partial f}{\partial x^i}$ denotes the partial derivative of the smooth function $f$ along the $i$th coordinate, and where we use the Einstein summation convention.

For emphasis we write

for the pointwise product of smooth functions.

Some examples of star products as in def. 123:

(…)

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Free quantum fields

We discuss here the quantum observables for the special case of free field theories (def. 62). In perturbative quantum field theory this is the basis of the construction of all interacting theories in the infinitesimal neighbourhood of the free field theories.

• Wick algebra and normal ordered products

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Wick algebra and normal ordered products

• Wick algebra and normal-ordered product

  $\mathcal{F}_{loc} \overset{:-:}{\longrightarrow} \mathcal{F}_{mc}[ [ \hbar ] ]$

(…)

To warm up, we first discuss how the star product (def. 123) of a finite dimensional vector space equipped with almost Kähler structure may be interpreted as “normal-ordered product for a single mode”:

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Now to generalize this to the infinite dimensional case of free field theory:

(After deformation quantization below, the distributions appearing in def. 126 are the origin of “operator-valued distributions” in perturbative quantum field theory).

This shows that microcausality in this case is related to conservation of momentum in th point interaction.

More generally:

In Minkowski spacetime the Hadamard state is simply the usual vacuum state $\vert vac \rangle$, hence the Hadamard distribution is, as a generalized function

Therefore the abstractly defined Wick algebra as in def. 127 in this case satisfies the relation

This is the traditional expression for the normal ordered Wick product on Minkowski spacetime (e.g. here).

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Scattering

We consider now the axioms for a perturbative S-matrix of a Lagrangian field theory as used in causal perturbation theory (def. 128 below). Since, by definition, the S-matrix is a formal sum of multi-linear continuous functionals, it is convenient to impose axioms on these directly: this is the axiomatics for time-ordered products in def. 129 below. That these latter axioms already imply the former is the statement of prop. 91 below. Its proof requires a close look at the “reverse-time ordered products” for the inverse S-matrix (def. 131 below) and their induced reverse-causal factorization (prop. 90 below).

The axioms we consider here are just the bare minimum of causal perturbation theory, sufficient to imply that the induced perturbative quantum observables organize into a causally local net of quantum observables (discussed below).

In applications one considers further axioms, in particular compatibility of the S-matrix with spacetime symmetry. This is needed for the proof of the main theorem of perturbative renormalization (see below).

Def. 128 is due to (Epstein-Glaser 73 (1)) (in view of prop. 92 below), except that these authors remain a little vague on the nature of the domain. The domain $\mathcal{F}_{loc}$ is made explicit (in terms of axioms for the time-ordered products, see def. 129 below), in (Brunetti-Fredenhagen 99, section 3, DütschFredenhagen 04, appendix E, Hollands-Wald 04, around (20)); for review see (Rejzner 16, around def. 6.7).

Definition 128 suggests to focus on the multilinear operations $T(...)$ which define the perturbative S-matix order-by-order:

This condensed notation turns out to be greatly simplify computations, as it absorbs all the “relative” combinatorial prefactors:

In order to prove that the axioms for time-ordered products do imply those for a perturbative S-matrix (prop. 91 below) we need to consider the corresponding reverse-time ordere products:

(e.g. Epstein-Glaser 73 (11))

In fact prop. 88 is a special case of the following more general statement:

(Epstein-Glaser 73, around (15))

Using these facts about the reverse-time ordered products, we may finally prove that time-ordered products indeed do induced a perturbative S-matrix:

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Quantum observables

A genuine local observable should depend on the values of the fields on some compact subset of spacetime. Moreover, a perturbative quantum observable should be a power series in Planck's constant $\hbar$, reducing to the corresponding classical observable at $\hbar = 0$. The perturbative S-matrix axiomatized above is neither localized in spacetime this way, nor is it a power series in $\hbar$ (it is a Laurent series in $\hbar$). So it is not a local observable. But the actual quantum observables on interacting fields may be expressed in terms of the S-matrix by Bogoliubov's formula (def. 132 below).

This formula is consistent in that it implies that local observables form a causally local net as their spacetime support varies (this is prop. 94 below). (On deeper grounds, this formula turns out to yield the formal Fedosov deformation quantization of the interacting field theory (Collini 16).)

Namely a key consequence of the “causal additivity” axiom on the S-matrix in def. 128 turns out to be that the perturbative quantum observables on interacting fields with compact spacetime support (def. 132)

1. depend on the adiabatic switching $g_{sw}$ of the interaction Lagrangian density only up to canonical unitary isomorphism (prop. 92 below)

2. form a causally local net of observables in the sense of the Haag-Kastler axioms as the spacetime localization varies (prop. 94 below).

To the extent that a local net of observables may be regarded as defining a quantum field theory, which is the claim of (perturbative) AQFT, this proves that the perturbative S-matrices in causal perturbation theory as in def. 128 indeed make sense, despite the involvement of adiabatic switching of the interaction Lagrangian density which does not make physical sense when interpreted naively: In reality the interaction is of course not (for realistic theories at least) “switched off” outside some bounded region of spacetime; but the result here shows that if we pretend that it does then first of all we get consistent mathematical formulas and moreover we can then nevertheless compute the correct quantum observables that are localized in this spacetime region. But the local net of observables as the spacetime localization varies is supposed to encode the full quantum field theory. Certainly any given experiment in practice probes a bounded spacetime region, and hence the algebra of observables localized in this region is sufficient to compare the theory to experiment.

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This definition of $\widehat{A}$ without the adiabatic switching $g_{sw}$ is originally due to Bogoliubov-Shirkov 59heory#BogoliubovShirkov59), nowadays sometimes called Bogoliubov's formula (e.g. Rejzner 16 (6.12)). The version with adiabatic switching is due to (Epstein-Glaser 73 around (74)). Review includes (Dütsch-Fredenhagen 00, around (17)).

The power series coefficients of the quantum observables on interacting fields are also called the retarded products. For the time being we mention these here just for completeness:

Direct axiomatization of the retarded products is due to (Dütsch-Fredenhagen 04), see (Collini 16, section 2.2).

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It is useful now to reformulate the causal additivity-property of the perturbative S-matrix in terms of the generating functions / retarded products:

We now use this fact (prop. 92) to neatly organize the system of localized quantum observables on interacting fields:

(Brunetti-Fredenhagen 99, (65)-(67))

(Dütsch-Fredenhagen 00, section 3, following Brunetti-Fredenhagen 99, section 8, Il’in-Slavnov 78)

In the same kind of way as def. 134 the actual net of algebra of perturbative quantum observables (def. 132) is defined:

As a corollary of prop. 93 we then have the key result:

(Dütsch-Fredenhagen 00, below (17), following Brunetti-Fredenhagen 99, section 8, Il’in-Slavnov 78)

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Feynman diagrams

So far we considered only the axioms on a consistent perturbative S-matrix /time-ordered products and its formal consequences. Now we discuss the actual construction of time-ordered products, hence of perturbative S-matrices, by the process called renormalization of Feynman diagrams.

We first discuss how time-ordered product, and hence the perturbative S-matrix above, is uniquely determined away from the locus where interaction points coincide (prop. 95 below). Moreover, we discuss how on that locus the time-ordered product is naturally expressed as a sum of products of distributions of Feynman propagators that are labeled by Feynman diagrams: the Feynman perturbation series (prop. 96 below).

This means that the full time-ordered product is an extension of distributions of these scattering amplitudes- to the locus of coinciding vertices. The space of possible such extensions turns out to be finite-dimensional in each order of $g/\hbar, j/\hbar$, parameterizing the choice of point-supported distributions at the interaction points whose scaling degree is bounded by the given Feynman propagators.

It follows that the problem of constructing time-ordered products, and hence (by prop. 91) the perturbative S-matrix, consists of finding compatible extension of the distribution $prod \circ \exp\left( \left\langle \omega_F , \frac{\delta }{\delta \phi} \otimes \frac{\delta}{\delta \phi} \right\rangle \right)$ to the diagonal.

Moreover, by the nature of the exponential expression, this means in each order to extend products of Feynman propagators labeled by graphs whose vertices correspond to the polynomial factors in $F$ and $G$ and whose edges indicate over which variables the Feynman propagators are to be multiplied.

(Keller 10, IV.1)

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Renormalization

renormalization

• extension of distributions

• main theorem of perturbative renormalization

…

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Quantum Electrodynamics

• quantum electrodynamics

• anomalous magnetic moment of the electron

(…)

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